Messages posted by tarinbansal

The half life for radio active decay of C-14 is 5730 years. An archaeological artefact contained wood that had only 80% of the C-14 found in a living tree. Estimate the age of the sample.

Is there any method to solve stoichiometry questions without knowing the reactions? If there is no such method, please advise me how to solve problems without knowing chemical reactions.

Hey its easy.

I = V/R (1-e^-Rt/L)

Put the values of I, V, R, and find the time t. You will get the answer.

The force is q(VxB) It means that v is perpendicular to the force. Therefore the work done is zero. It means that change in kinetic energy is zero. Hence there is no change in the velocity of the particle.

Yes the path is of uniform radius provided the magnetic field is constant in that region.

Yes . R=mv/qb. If R(radius) is within the range of the field it will complete the circle.

Rate me if this solves your doubts.

Fortunately or unfortunately, competitive exams have become a strict norm for selection to various professional courses in our country. Notwithstanding the demerits of such a system, competitive exams are a reality for the lakhs of students who appear in them every year.

In such a scenario, everyone wants to success. Success in these exams brings immense glory and confidence to the individuals, besides giving them a clear cut break into the career line of their choice. However, seats are limited and not everyone who appears can be selected. This does not mean that those who are not selected do not deserve a training in these courses. It only means that there aren't sufficient seats to accommodate all the deserving candidates.

Keeping this background in mind, let us explore what it takes to succeed in these competitive exams particularly at the undergraduate level.

To my mind, the first prerequisite for success in these examinations is determination. The candidate must be quite sure of his level of motivation. Without a high level of motivation, no one can go through the rigorous preparation process. Thus, you must be absolutely determined to give your best efforts in your endeavor.

Once one is sufficiently motivated, one needs to be ready to give sufficient number of hours of his or her time in studies. This is real hard work and involves significant sacrifices on the part of the individual. "All work and no play, makes Jack a dull boy". Therefore it is also important that the candidate does take out some time for relaxation either through sports, music or other art forms. Regular exercise ( about 15 - 20 minutes ) every day will help maintain a healthy body which in turn will house a healthy mind.

The third requirement, is selection of the right study material. One should not go for shortcuts. Instead, try to understand the chapters / concepts in detail and practice sufficient number number of questions. Therefore go for standard textbooks, study materials provided, as various correspondence courses and other coaching institutions serve as supplementary and complementary material to the textbook material.

The fourth consideration is an appropriate 'Study Strategy'. A very common question which students ask is, How to balance the school studies with competitive exam study. T think that school studies can be incorporated within the broad framework of one's competitive exams' preparation strategy. One strategy is to look at each chapter as a chapter to be learnt, a material to be mastered; rather than dichotomizing it as school & competitive exam-related study material. Ultimately, if one's clear about the concepts of say, "Current & Electricity", and has practiced sufficient number of questions / numericals ; one can face any examination whether of school or competition. One can use periodical tests in school as an opportunity to master the material that one has collected rather than restricting yourself to the limited knowledge & practice that is needed for doing well in the periodical test itself. Of course, you may not be able to complete all the practice material for one test but clear the concepts. You can use the rest of the practice material for revision and self evaluation at a later time. As the examinations come closer, practicing previous years papers is also key to doing well in these examinations.

Also revise the important concepts, facts, problems that you would have marked during during your 1.5 - 2 years of preparation before the examination.

On the day of examination, keep your cool and attempt the questions. Remember, if you have given your best during the last 1.5 - 2 years, you need not fear anything. Ultimately, only effort and not the result is in your hand. Which you should do to your maximum capability as outlined above. Rest is your destiny or luck. So go ahead, give your best and leave the rest!

Happy preparing!!

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Hey its fabulous

Vaastu tips to help you study better
vaastu -- what direction in right from me ??

Many a times students complain that they "can't remember the lesson" or "our retention power is weak" or "we can't concentrate on our studies" etc. The remedy to these problems was formulated by our saints in the olden times, in the form of the secretive science of Vaastu Shastra. For students eager to do well, here are a few techniques, following which the students can achieve success in their studies and obviously bring about a positive change in their grades.

The room to be chosen for study should be in the East, North or the North-East direction of the house.
These directions improve the absorption power and increase the knowledge content. Along with that if the door of the room is also in either of these directions, the result turns out to be the best.
Learners should face East or North in order to memorize quickly and accelerate retention as well as concentration power.
The study room should have images of Lord Ganesha and Goddess Saraswati.
The books in the study chamber should be kept in the South-West, South or West direction and not in the North-East. In addition to that if the North-East direction bears heavy weight it shall pressurize the student's mind.
Furthermore, if the student studies on the table and chair, the table should not stick to the wall. It should be at least 2-3 feet away from it. Along with that unnecessary books shouldn't be piled up on the table. Only the books which are needed should be kept on the table. Loads of books on the table create unnecessary mental pressure on the mind. The study books should be placed in the cupboard or the cabinet rather than in the open. And if the shutter of the cabinet or cupboard is kept shut, the flow of energy is constantly maintained.
If the learner makes use of a table lamp while studying then the lamp should be kept in the South-East corner of the desk. Plus if it is a square table then nothing like it. The study table and the chair must not be facing the door. Further the size of the study table should neither be too big nor too small. It should be just the size which is comfortable for the working of the student. A big sized table diminishes the working capacity, whereas a small table gives rise to depression.
Moreover, if possible, the walls of the study room should be light in colour as it is very auspicious and enhances the wishing power as well as supports the speedy progress of mind.
Along with that, at the time of study, the learner should use the pyramid cape as it proves to be extremely beneficial. Not only that it embellishes the remembering power and absorption.
The student should ensure that after studying, when they go off to sleep, their head is in the South direction. It helps in maintaining the magnetic balance of the body and the earth.
And last but not the least, a watch is a must in each and every study room.

So, if you keep in mind these easy techniques you shall note a drastic upward shift in you grades. With hard work, of course. Keep studying hard but remember in the right direction, as Vaastu Shastra is nothing but the science of directions.

PLEASE LEAVE UR VALUABLE COMMENTS AND RATE IF THIS PROVES TO BE USEFUL.

Geomagnetic field- FAQs

Visit

http://sol.sci.uop.edu/~jfalward/electricpotential/electricpotential.html

Believe me, its a very good site. You will find some excellent facts about equipotential surfaces.

In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. In this article, the entries of a matrix are real or complex numbers unless otherwise noted.

Matrices are useful to record data that depend on two categories, and to keep track of the coefficients of systems of linear equations and linear transformations

Definitions and notations

The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (or m×n matrix) and m and n are called its dimensions.

The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A. This is written as A_i,j or A[i,j].

We often write $A:=(a_{i,j})_{m imes n}$ to define an m × n matrix A with each entry in the matrix A[i,j] called a_ij for all 1 ? i ? m and 1 ? j ? n.

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Example

The matrix

$egin{bmatrix} 1 & 2 & 3 \ 1 & 2 & 7 \ 4&9&2 \ 6&0&5end{bmatrix}$

is a 4×3 matrix. The element A[2,3] or a_2,3 is 7.

[edit]

Adding, subtracting, and multiplying matrices

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Sum

Main article: Matrix addition

If two m-by-n matrices A and B are given, we may define their sum A + B as the m-by-n matrix computed by adding corresponding elements, i.e., (A + B)[i, j] = A[i, j] + B[i, j]. For example

$egin{bmatrix} 1 & 3 & 2 \ 1 & 0 & 0 \ 1 & 2 & 2 end{bmatrix} + egin{bmatrix} 0 & 0 & 5 \ 7 & 5 & 0 \ 2 & 1 & 1 end{bmatrix} = egin{bmatrix} 1+0 & 3+0 & 2+5 \ 1+7 & 0+5 & 0+0 \ 1+2 & 2+1 & 2+1 end{bmatrix} = egin{bmatrix} 1 & 3 & 7 \ 8 & 5 & 0 \ 3 & 3 & 3 end{bmatrix}$

Another, much less often used notion of matrix addition is the direct sum.

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Difference

Main article: Matrix subtraction

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Scalar multiplication

If a matrix A and a number c are given, we may define the scalar multiplication cA by (cA)[i, j] = cA[i, j]. For example

$2 egin{bmatrix} 1 & 8 & -3 \ 4 & -2 & 5 end{bmatrix} = egin{bmatrix} 2 imes 1 & 2 imes 8 & 2 imes -3 \ 2 imes 4 & 2 imes -2 & 2 imes 5 end{bmatrix} = egin{bmatrix} 2 & 16 & -6 \ 8 & -4 & 10 end{bmatrix}$

These two operations turn the set M(m, n, R) of all m-by-n matrices with real entries into a real vector space of dimension mn.

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Multiplication

Main article: Matrix multiplication

Multiplication of two matrices is well-defined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an m-by-n matrix (m rows, n columns) and B is an n-by-p matrix (n rows, p columns), then their product AB is the m-by-p matrix (m rows, p columns) given by

(AB)[i, j] = A[i, 1] * B[1, j] + A[i, 2] * B[2, j] + ... + A[i, n] * B[n, j] for each pair i and j.

For instance:

$egin{bmatrix} 1 & 0 & 2 \ -1 & 3 & 1 \ end{bmatrix} imes egin{bmatrix} 3 & 1 \ 2 & 1 \ 1 & 0 end{bmatrix} = egin{bmatrix} (1 imes 3 + 0 imes 2 + 2 imes 1) & (1 imes 1 + 0 imes 1 + 2 imes 0) \ (-1 imes 3 + 3 imes 2 + 1 imes 1) & (-1 imes 1 + 3 imes 1 + 1 imes 0) \ end{bmatrix} = egin{bmatrix} 5 & 1 \ 4 & 2 \ end{bmatrix}$

This multiplication has the following properties:

(AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity").
(A + B)C = AC + BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity").
C(A + B) = CA + CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity").

It is important to note that commutativity does not generally hold; that is, given matrices A and B and their product defined, then generally AB ? BA.

Matrices are said to anticommute if AB = -BA. Such matrices are very important in representations of Lie algebras and in Representations of Clifford algebras

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Linear transformations, ranks and transpose

Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next. This same property makes them powerful data structures in high-level programming languages.

Here and in the sequel we identify Rⁿ with the set of "rows" or n-by-1 matrices. For every linear map f : Rⁿ -> R^m there exists a unique m-by-n matrix A such that f(x) = Ax for all x in Rⁿ. We say that the matrix A "represents" the linear map f. Now if the k-by-m matrix B represents another linear map g : R^m -> R^k, then the linear map g o f is represented by BA. This follows from the above-mentioned associativity of matrix multiplication.

More generally, a linear map from an n-dimensional vector space to an m-dimensional vector space is represented by an m-by-n matrix, provided that bases have been chosen for each.

The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.

The transpose of an m-by-n matrix A is the n-by-m matrix A^tr (also sometimes written as A^T or ^tA) formed by turning rows into columns and columns into rows, i.e. A^tr[i, j] = A[j, i] for all indices i and j. If A describes a linear map with respect to two bases, then the matrix A^tr describes the transpose of the linear map with respect to the dual bases, see dual space.

We have (A + B)^tr = A^tr + B^tr and (AB)^tr = B^tr * A^tr.

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Square matrices and related definitions

A square matrix is a matrix which has the same number of rows as columns. The set of all square n-by-n matrices, together with matrix addition and matrix multiplication is a ring. Unless n = 1, this ring is not commutative.

M(n, R), the ring of real square matrices, is a real unitary associative algebra. M(n, C), the ring of complex square matrices, is a complex associative algebra.

The unit matrix or identity matrix I_n, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI_n=M and I_nN=N for any m-by-n matrix M and n-by-k matrix N. For example, if n = 3:

$I_3 = egin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{bmatrix}$

The identity matrix is the identity element in the ring of square matrices.

Invertible elements in this ring are called invertible matrices or non-singular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that

AB = I_n ( = BA).

In this case, B is the inverse matrix of A, denoted by A^?1. The set of all invertible n-by-n matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group.

If ? is a number and v is a non-zero vector such that Av = ?v, then we call v an eigenvector of A and ? the associated eigenvalue. (Eigen means "own" in German.) The number ? is an eigenvalue of A if and only if A??I_n is not invertible, which happens if and only if p_A(?) = 0. Here p_A(x) is the characteristic polynomial of A. This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In this sense, every square matrix has n complex eigenvalues.

The determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Leibniz formula. Invertible matrices are precisely those matrices with nonzero determinant.

The Gauss-Jordan elimination algorithm is of central importance: it can be used to compute determinants, ranks and inverses of matrices and to solve systems of linear equations.

The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.

Every orthogonal matrix is a square matrix.

Matrix exponential is defined for square matrices, using power series.

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Special types of matrices

In many areas in mathematics, matrices with certain structure arise. A few important examples are

Symmetric matrices are such that elements symmetric to the main diagonal (from the upper left to the lower right) are equal, that is, a_i,j=a_j,i.
Skew-symmetric matrices are such that elements symmetric to the main diagonal are the negative of each other, that is, a_i,j= - a_j,i. In a skew-symmetric matrix, all diagonal elements are zero, that is, a_i,i=0.
Hermitian (or self-adjoint) matrices are such that elements symmetric to the diagonal are each others complex conjugates, that is, a_i,j=a^*_j,i, where the superscript '*' signifies complex conjugation.
Toeplitz matrices have common elements on their diagonals, that is, a_i,j=a_i+1,j+1.
Stochastic matrices are square matrices whose columns are probability vectors; they are used to define Markov chains.

For a more extensive list see list of matrices.

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Matrices in abstract algebra

If we start with a ring R, we can consider the set M(m,n, R) of all m by n matrices with entries in R. Addition and multiplication of these matrices can be defined as in the case of real or complex matrices (see below). The set M(n, R) of all square n by n matrices over R is a ring in its own right, isomorphic to the endomorphism ring of the left R-module Rⁿ.

Similarly, if the entries are taken from a semiring S, matrix addition and multiplication can still be defined as usual. The set of all square n×n matrices over S is itself a semiring. Note that fast matrix multiplication algorithms such as the Strassen algorithm generally only apply to matrices over rings and will not work for matrices over semirings that are not rings.

If R is a commutative ring, then M(n, R) is a unitary associative algebra over R. It is then also meaningful to define the determinant of square matrices using the Leibniz formula; a matrix is invertible if and only if its determinant is invertible in R.

All statements mentioned in this articles for real or complex matrices remain correct for matrices over an arbitrary field.

Matrices over a polynomial ring are important in the study of control theory.

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In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.

For a fixed positive integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In particular, this is true when R is the field of real or complex numbers.

A determinant of A is also sometimes denoted by |A|, but this notation is ambiguous: it is also used to for certain matrix norms, and for the square root of

A A *

.

Determinants of 2-by-2 matrices

The 2×2 matrix

$A=egin{bmatrix}a&b\ c&dend{bmatrix}$

has determinant

$det(A)=ad-bc ,$ .

The interpretation when the matrix has real number entries is that this gives the area of the parallelogram with vertices at (0,0), (a,c), (b,d), and (a + b, c + d), with a sign factor (which is ?1 if A as a transformation matrix flips the unit square over).

A formula for larger matrices will be given below .

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Applications

Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix

A

through the characteristic polynomial

$p(x) = det(xI - A) ,$

where I is the identity matrix of the same format as A.

One often thinks of the determinant as assigning a number to every sequence of

n

vectors in $Bbb{R}^n$ , by using the square matrix whose columns are the given vectors. With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation in Euclidean spaces. The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed.

Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map $f: Bbb{R}^nightarrow Bbb{R}^n$ is represented by the matrix

A

, and

S

is any measurable subset of $Bbb{R}^n$ , then the volume of

f (S)

is given by $left| det(A)ight| imes operatorname{volume}(S)$ . More generally, if the linear map $f: Bbb{R}^nightarrow Bbb{R}^m$ is represented by the

m

-by-

n

matrix

A

, and

S

is any measurable subset of $Bbb{R}^{n}$ , then the

n

-dimensional volume of

f (S)

is given by $sqrt{det(A^ op A)} imes operatorname{volume}(S)$ . By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines.

The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a?b, b?c, c?d)|, or any other combination of pairs of vertices that form a simply connected graph.

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General definition and computation

Suppose $A = (A_{i,j}) ,$ is a square matrix.

If

A

is a 1-by-1 matrix, then $det(A) = A_{1,1} ,$

If

A

is a 2-by-2 matrix, then $det(A) = A_{1,1}A_{2,2} - A_{2,1}A_{1,2} ,$

For a 3-by-3 matrix

A

, the formula is more complicated:

$egin{matrix} det(A) & = & A_{1,1}A_{2,2}A_{3,3} + A_{1,3}A_{2,1}A_{3,2} + A_{1,2}A_{2,3}A_{3,1}\ & & - A_{1,3}A_{2,2}A_{3,1} - A_{1,1}A_{2,3}A_{3,2} - A_{1,2}A_{2,1}A_{3,3}. end{matrix},$

For a general

n

-by-

n

matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:

$det(A) = sum_{sigma in S_n} sgn(sigma) prod_{i=1}^n A_{i, sigma(i)}$

The sum is computed over all permutations

?

of the numbers {1,2,...,n} and

sgn(?)

denotes the signature of the permutation

?

: +1 if

?

is an even permutation and ?1 if it is odd (see even and odd permutations).

This formula contains

n!

(factorial) summands and is therefore impractical to use it to calculate determinants for large

n

.

In general, determinants can be computed with the Gauss algorithm using the following rules:

If $A$ is a triangular matrix, i.e. $A_{i,j} = 0 ,$ whenever $i > j$ , then $det(A) = A_{1,1} A_{2,2} cdots A_{n,n} ,$
If $B$ results from $A$ by exchanging two rows or columns, then $det(B) = -det(A) ,$
If $B$ results from $A$ by multiplying one row or column with the number $c$ , then $det(B) = c,det(A) ,$
If $B$ results from $A$ by adding a multiple of one row to another row, or a multiple of one column to another column, then $det(B) = det(A) ,$

Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.

It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices. To do this along row

i

, say, we write

$det(A) = sum_{j=1}^n A_{i,j}C_{i,j} = sum_{j=1}^n A_{i,j} (-1)^{i+j} M_{i,j}$

where the

C i, j

represent the matrix cofactors, i.e.

C i, j

is

( ? 1) i + j

times the minor

M i, j

, which is the determinant of the matrix that results from

A

by removing the

i

-th row and the

j

-th column.

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Example

Suppose we want to compute the determinant of

$A = egin{bmatrix}-2&2&-3\ -1& 1& 3\ 2 &0 &-1end{bmatrix}$

We can go ahead and use the Leibniz formula directly:

$det(A),$	$=,$	$(-2)cdot 1 cdot (-1) + (-3)cdot 0 cdot (-1) + 2cdot 3cdot 2$
		$- (-3)cdot 1 cdot 2 - (-2)cdot 3 cdot 0 - 2cdot (-1) cdot (-1)$
	$=,$	$2 + 0 + 12 - (-6) - 0 - 2 = 18.,$

Alternatively, we can use Laplace's formula to expand the determinant along a row or column. It is best to choose a row or column with many zeros, so we will expand along the second column:

$det(A),$	$=,$	$(-1)^{1+2}cdot 2 cdot det egin{bmatrix}-1&3\ 2 &-1end{bmatrix} + (-1)^{2+2}cdot 1 cdot det egin{bmatrix}-2&-3\ 2&-1end{bmatrix}$
	$=,$	$(-2)cdot((-1)cdot(-1)-2cdot3)+1cdot((-2)cdot(-1)-2cdot(-3))$
	$=,$	$(-2)(-5)+8 = 18.,$

A third way (and the method of choice for larger matrices) would involve the Gauss algorithm. When doing computations by hand, one can often shorten things dramatically by smartly adding multiples of columns or rows to other columns or rows; this doesn't change the value of the determinant, but may create zero entries which simplifies the subsequent calculations. In our example, adding the second column to the first one is especially useful:

$egin{bmatrix}0&2&-3\ 0 &1 &3\ 2 &0 &-1end{bmatrix}$

and this determinant can be quickly expanded along the first column:

$det(A),$	$=,$	$(-1)^{3+1}cdot 2cdot det egin{bmatrix}2&-3\ 1&3end{bmatrix}$
	$=,$	$2cdot(2cdot3-1cdot(-3)) = 2cdot 9 = 18.,$

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Properties

The determinant is a multiplicative map in the sense that

$det(AB) = det(A)det(B) ,$ for all n-by-n matrices

A

and

B

.

This is generalized by the Cauchy-Binet formula to products of non-square matrices.

It is easy to see that $det(rI_n) = r^n ,$ and thus

$det(rA) = det(rI_n cdot A) = r^n det(A) ,$ for all

n

-by-

n

matrices

A

and all scalars

r

.

The matrix

A

(over the real or complex numbers, or some other field) is invertible if and only if det(A)?0; in this case we have

$det(A^{-1}) = det(A)^{-1} ,$

Expressed differently: the vectors v₁,...,v_n in Rⁿ form a basis if and only if det(v₁,...,v_n) is non-zero.

A real matrix and its transpose have the same determinant:

$det(A^ op) = det(A) ,$ .

The determinants of a complex matrix and of its conjugate transpose are conjugate:

$det(A^*) = det(A)^* ,$ .

(Note the conjugate transpose is identical to the transpose for a real matrix)

If

A

and

B

are similar, i.e., if there exists an invertible matrix

X

such that

A

=

X ? 1 B X

, then by the multiplicative property,

$det(A) = det(B) ,$

This means that the determinant is a similarity invariant. Because of this, the determinant of some linear transformation T : V ? V for some finite dimensional vector space V is independent of the basis for V. The relationship is one-way, however: there exist matrices which have the same determinant but are not similar.

If

A

is a square

n

-by-

n

matrix with real or complex entries and if ?₁,...,?_n are the (complex) eigenvalues of

A

listed according to their algebraic multiplicities, then

$det(A) = lambda_{1}lambda_{2} cdots lambda_{n}$

This follows from the fact that

A

is always similar to its Jordan normal form, an upper triangular matrix with the eigenvalues on the main diagonal.

From this connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:

$det(exp(A)) = exp(operatorname{tr}(A))$ .

Performing the substitution $A mapsto ln A$ in the above equation yields

$det(A) = e^{mbox{tr}(ln A)}. \$

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Derivative

The determinant of real square matrices is a polynomial function from $Bbb{R}^{n imes n}$ to $Bbb{R}$ , and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

$d ,det(A) = operatorname{tr}(operatorname{adj}(A) ,dA)$

where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have

$d ,det(A) = det(A) ,operatorname{tr}(A^{-1} ,dA)$

or, more colloquially,

$det(A + X) - det(A) approx det(A) ,operatorname{tr}(A^{-1} X)$

if the entries in the matrix

X

are sufficiently small. The special case where

A

is equal to the identity matrix

I

yields

$det(I + X) approx 1 + operatorname{tr}(X)$ .

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Generalizations and related functions

As was pointed out above, it is possible to unambiguously define the determinant of any linear map f : V ? V, if V is a finite-dimensional vector space.

It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix

A

is invertible if and only if

det(A)

is an invertible element of the ground ring.

Abstractly, one may define the determinant as a certain anti-symmetric multilinear map as follows: if

R

is a commutative ring and

M = R n

denotes the free R-module with

n

generators, then

$det: M^nightarrow R$

is the unique map with the following properties:

det is $R$ -linear in each of the $n$ arguments.
det is anti-symmetric, meaning that if two of the $n$ arguments are equal, then the determinant is zero.
$det(e_1,ldots,e_n) = 1$ , where $e i$ is that element of $M$ which has a 1 in the $i$ -th coordinate and zeros elsewhere.

Linear algebraists prefer to use the multilinear map approach to define determinant, whereas combinatorialists may prefer the Leibniz formula. (Of course, even when using the above abstract approach, one has to use the Leibniz formula to show that such a multilinear map actually exists.)

The Pfaffian is an analog of the determinant for $2n imes 2n$ antisymmetric matrices. It is a polynomial of degree

n

, and its square is equal to the determinant of the matrix.

There is no direct generalisation of determinants, or of the notion of volume, to spaces of infinite dimension. There are various approaches possible, including the use of the extension of the trace of a matrix, and functional determinants.

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Do it with the help of straight lines. Draw the triangle on the cartesian plane. The area of triangle is 1/2( Determinant value of its vertices).
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The formula of hybridisation you used is wrong.
The formula is h=1/2(v+m-c+a).

We say that work done by magnetic field is zero. But when a dipole is placed at an angle to the magnetic dipole, it is rotated in the direction of magnetic field.

Suppose the magnet was at rest. Now when it moves it gains kinetic energy. Now change in kinetic energy=work done.Therefore some work is done in moving the magnet. Who does this work if the work done by magnetic field is zero?

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Demonstrate that a light beam reflected from three mutually perpendicular plane mirrors in succession reverses its direction.

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Definitions and notations

Example

Adding, subtracting, and multiplying matrices

Sum

Difference

Scalar multiplication

Multiplication

Linear transformations, ranks and transpose

Square matrices and related definitions

Special types of matrices

Matrices in abstract algebra

Determinants of 2-by-2 matrices

Applications

General definition and computation

Example

Properties

Derivative

Generalizations and related functions

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