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Algebra
16 Apr 2007 11:37:13 IST
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SHORT NOTES & MCQ'S...............MATRICES & DETERMINANTS
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SHORT NOTES: Def: A matrix is a rectangular array of numbers ( real or complex). 1. A matrix A = (aij)m ´ n over the field of complex numbers is said to be a rectangular matrix if m ? n. a square matrix if m = n. a row matrix if n = 1 a column matrix if n = 1 a null matrix if aij = 0 for all i, j. a diagonal matrix if m = n and aij = 0 for i? j a scalar matrix if m = n and aij = 0 for i ? j and a11 = a22 = a33 = ??. = ann Unit (or) identity matrix if m = n, aij = 0 for i ? j and aij = 1 for all i.2. (a) A square matrix A = (aij) n ´ n is an upper triangular matrix if aij = 0 for i > j A square matrix A = (aij) n ´ n is lower triangular matrix if aij = 0 for i < j. A square matrix A = (aij) n ´ n is symmetric if aij = aji for all i, j. A square matrix A = (aij) n ´ n is skew symmetric if aij = - aij for all i, j.3. Determinants To every square matrix A = [aij] n ´ n is associated a number or function called the determinant of A and is denoted by ½A½or detA. a11 a12 a13 If ½A½ = a21 a22 a23 = a11 (a22 a33 - a23 a32) - a12 (a21 a33 ? a23 a31) a31 a32 a33 +a13(a21 a32 ? a22 a31). ½A½ = ½A' ½or det A = A' . i.e. the value of a determinant remains unaltered if its rows and columns are interchanged. det(AB) = (det A) (det B) where A, B are square matrices of the same order.If A = (aij) n ´ n, n > 1 and B be the matrix obtained from A by interchanging two of its rows, or columns, then det B = -det A. If two rows(or columns) of a square matrix A are identical, then det A = 0. det B = K detA. If B is obtained from A by multiplying one row (or column) of A by K.Adj A = transpose of the matrix obtained from A by replacing each entry with its corresponding cofactor. A(adj A) = (AdjA) A = ½A½I. A square matrix A is non ? singular if ½A½ ? 0 and is singular if ½A½= 0.A square matrix A is said to be invertible if $ a square matrix B such that AB = BA = I. Here B = A-1 or A = B-1 A ?1 = (adj A)
½A½
Cramer?s Rule.
If Ax = B, det A ? 0, then x = D1 ,
D
y = D2 , z = D3 etc. where D = det a.
D D
Orthogonal Matrix:
A square matrix A is called an orthogonal matrix if AA-1 = I.
Nilpotent Matrix:
A square matrix A for which Ap = o, where p is a positive integer is called nilpotent matrix.
Idempotent matrix;
If A is a square matrix such that A2 = A, then A is called idempotent.
Involutary matrix:
A square matrix A is said to be involutary matrix if A2 = I
Rank of matrix:
A non ? zero matrix A is said to have rank r if there exists at least one minor of order r which is non ? zero and every minor of order ( r + 1) is zero.
A square matrix of order n is non singular if its rank r = n, i.e if |A | ? 0, then rank of
A = n
Rank of a matrix A is denoted by = ?(A)
Some properties
1. If ?(A) = ?(B) = n, then ?(AB) = n (where A, B are square matrices of order n).
2. The rank of a matrix whose every element is unity is 1.
3. Every skew symmetric matrix of odd order has rank less than its order.
4. The rank of the null matrix is not defined and the rank of every non ? null matrix is
greater than or equal to 1.
5. If A is an m ´ n matrix, then ?(A) ? Min (m,n).
6. ?(A) = ?(A' )
7. Elementary transformation do not alter the rank of a matrix..
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Blazing goIITian
Joined: 26 Feb 2007
Posts: 373
16 Apr 2007 11:39:36 IST
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Multiple Choice Questions
1. The transformation due to reflection of (x,y) through the origin is described by the matrix.
(a) 0 0 (b) ?1 0 (c) 0 -1 (d) 1 0
-1 1 0 -1 -1 0 0 1
2 The point (3,2) is reflected in the y ? axis and then moved a distance of 5 units towards the negative side of y- axis. The coordinates of the point thus obtained are
(a) (-3, -3) (b) (3,3) (c)-3, 3) (d) (3, -3).
If A and B are two matrices such that AB = BA, then for every n Î N
(a)A n B = AnB (b) (AB)n = An Bn
(c)(A + B)n = nC0An + nC1An-1B + nC2An-2B2 + ?.. +nCn Bn
(d)A2n-B2n = (An-Bn)(An +Bn)
If a, b, c are in A.P. then the value of
x +1 x + 2 x + a
x + 2 x + 3 x + b is:
x + 3 x + 4 x + c
(a) 3 (b) ?3 (c)0 (d) none of these
The value of the determinant
a2 a 1
D = cos (nx) cos(n+ 1)x cos(n+2)x is independent of
sin (nx) sin(n + 1)x sin(n+2)x
(a) n (b)a ( c)x (d) none of these
The value of x for which the matrix product
2 0 7 -x 14x 7x
0 1 0 0 1 0 equals an identity matrix is
1 -2 1 x -4x -2x
(a) 1 (b) 1 (c) 1 (d) 1
2 3 4 5
7 If a ¹ b ¹ c , one value of x which satisfies the equation
0 x-a x-b
x+a 0 x- c = 0 is given by
x+b x+ c 0
(a)x = a (b) x = b ( c) x = c (d)x = 0
8 If A,B, C be three square matrices such that A = B+ C, then detA =
(a)detB + det C (b) det B ( c)det C (d) none of these
9 If A and B are arbitrary square matrices of the same order, then
(a) (AB)¢ = A¢ B¢ (b) (A¢)¢ (B¢)¢ = B¢ A¢ ( c) (A + B) ¢ = A¢- B¢ (d) (AB)¢ = B¢ A¢
10 If A and B are two matrices such that A + B and AB are both defined, then
(a) A and B can be any matrices
(b) A, B are square matrices not necessarily of same order4
(c) A, B are square matrices of same order
(d) number of columns of A = number of rows of B
11 If 1, w , w2 are the cube roots of unity , then
1 wn w2n
D = w2n 1 wn has the value
wn w2n 1
(a) 0 (b) w (c) w2 (d)1
cos a - sin a 0 cos b 0 sinb
12. Let F(a) sin a cos a 0 G ( b) = 0 1 0
0 0 1 - sin b 0 cos b
Then [ F ( a) G(b) ] ?1 is equal to
(a) F (a) ? G (b) (b) ? F(a)- G(b)
(c)[F(a)] ?1 [ G(b)]-1 (d) [ G ( b)] ?1 [ F(a)]-1
12 22 32 42
13 The value of determinant D = 22 32 42 52 is =
32 42 52 62
42 52 62 72
(a)1 (b)0 ( c)2 (d)3
a h g x
14 The order of [ x y z] h b f y is
g f c z
(a) 3 ´ 1 (b) 1´ 1 (c) 1´ 3 (d) 3 ´ 3
15 The system of linear equations x + y + z = 2, 2x + y ? z = 3 , 3x + 2y + kz = 4 has a unique solution if
(a) k ¹ 0 (b) ?1 < k <1 (c)-2 < k < 2 (d) k = 0
16 If a > 0 , b > 0, c > 0 are respectively the pth, qth, rth terms of a G.P., then the value of the determinant.
log a p 1
log b q 1 is
log c r 1
(a) 1 (b) 0 (c) ?1 (d) none of these
17 If A is a square matrix, then A-A¢ is a
(a) diagonal matrix (b) skew-symmetric matrix
(c)symmetric matrix (d) none of these.
18. If A and B are square matrices of order 3 such that | A | = -1 , | B | = 3 , then |3AB| = .. .
(a) ? 9 (b) ?81 (c) ?27 (d)81.
19 The value of ?a? for which the system of equations
a3x + (a + 1)3 y + (a +2)3z = 0 ; ax + (a + 1) y + (a + 2) z = 0; x + y + z = 0
has a non-zero solution is
(a) 1 (b)0 (c) ? 1 (d) none of these
1 a a2
20 cos(n-1)x cos nx cos(n + 1) x is independent of
sin (n-1) x sin x sin( n + 1) x
(a) x (b) a (c)n (d) none of these
21 If Dr = 2 r-1 2.3r-1 4.5 r-1 n
a b g , then the value of S Dr is
2n ? 1 3n ?1 5n ? 1 r = 1
(a) 0 (b) abg (c)a + b + g (d) a.2n + b.3n + g .4n
22 If a1, a2, a3 , ?.. form a G.P and ai > 0 for all i ³ 1, then
log a m log a m + 1 log a m + 2
D= log a m + 3 log a m+ 4 log a m+ 5 is equal to
log a m + 6 log a m+ 7 log a m + 8
(a)log a m + 8 ? log am (b) log a m + 8 + log am (c)0 (d) (log a m + 4)2
23 In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can decomposed into n determinants, where n has value.
(a) 1 (b) 9 (c)16 (d) 24
24 The number of distinct real roots of
sin x cos x cos x
cos x sin x cos x = 0 in the interval -p / 4 £ x £ p / 4 is
cos x cos x sin x
(a)0 (b)2 (c)1 (d)3
25 If A is an invertible matrix and B is a matrix, then
(a) rank (AB) = rank (A) (b) rank (AB) = rank (B)
(c)rank (AB) > rank (A) (d)rank (AB) > rank (B)
26 If A + B + C = p then value of
sin (A + B + C) sin B sin C
- sin B 0 tan A is
cos (A + B) -tan A 0
(a) 0 (b) 1 (c)2 sin B tan B cos C (d) none of these
27 If E (a ) = cos2 a cos asin a and a, b differ by an odd
cosa sin a sin2a
multiple of p/2 , then E (a). E (b) is a
(a) null matrix (b) unit matrix ( c) diagonal matrix (d) none of these
a -1 0
28 If f(x) = ax a -1 , then f (2x) ? f(x) =
ax2 ax a
(a) a(2a + 3x) (b)ax (2x+ 3a) ( c) ax (2 a + 3 x) (d) x (2 a + 3x)
29 If A is a square matrix of order n such that its elements are polynomials in x and its
r ? rows become identical for x = a, then
(a) (x-a)r is a factor of | A | (b) (x-a)r-1 is a factor of | A|
( c)(x-a)r +1 is a factor of | A | (d) )(x-a)r is a factor of A
30 If capital letters denotes the co factors of the corresponding small letters in
a1 b1 c1 A1 B1 C1
D = a2 b2 c2 , then D¢ = A2 B2 C2 is
a3 b3 c3 A3 B3 C3
(a)D (b)D2 ( c)2D (d)0
Reply
16 Apr 2007 11:50:09 IST
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MCQ
1. If A = i 0 ,n ? N, then A 4n equals
0 i
(a) 1 0 (b) i 0
0 1 0 i
(c) 0 i (d) 0 0
i 0 (d) 0 0
2. If a b is to be square root of the two rowed unit matrix, then a, b and
-a
g should satisfy the relation.
1 + a2 + b g= 0 (b) 1 ? a2 - b g = 0
(c) 1 - a2 + b g = 0 (d) 1 + a 2 - b g = 0
3. If w is a complex cube rot of unity, then the value of the determinant
1 w w +1
w + 1 1 w is ;
w w + 1 1
(a) 0 (b)w (c) 2 (d) 4
4. The value of the determinant
1! 2! 3!
D = 2! 3! 4! is :
3! 4! 5!
(a) 2! (b) 3! (c) 4! (d)5!
5. If [ ] denotes the greatest integer less than or equal to the real number under consideration, and ?1 ? x .< 0; 0 ? y <1, 1? z <2, then the value of the determinant.
[x] + 1 [y] [z]
[x] [y] + 1 [z] is ;
[x] [y] [z] + 1
(a) [x] (b) [y] (c) [z] (d) none of these
If A and B are square matrices of the same order and A is non singular then for a positive integer n, (A ?1 BA)n is equal to
(a)A ?n Bn An (b) An Bn A ?n
(c) A-1 Bn A (d) n (A-1 BA)
If a,b,c are non-zero real numbers, then the inverse of the matrix
a 0 0
A = 0 b 0 is :
0 0 c
a-1 0 0 a-1 0 0
(a) 0 b-1 0 (b) abc 0 b-1 0
0 0 c-1 0 0 c-1
(c) 1 1 0 0 a 0 0
abc 0 1 0 (d) 1 0 b 0
0 0 1 abc 0 0 c
If the entries in a 3 ´ 3 determinant are either 0 or 1, then the greatest value of this determinant is
(a) 1 (b) 2 (c) 3 (d) 9
a1 b1 c1 a1 + pb1 b1+ qc1 c1 + ra1
9. If D = a2 b2 c2 and D' = a2 + pb2 b2 + qc2 c2 + ra2 , then
a3 b3 c3 a3 + pb3 b3 + qc3 c3 + ra3
(a) D' = D (b) D' = D (1-pqr)
(c) D' = D (1 + p + q + r) (d) D' = D (1 + pqr)
If A is singular matrix, then Adj. A is
(a) Singular (b) non-singular (c) symmetric (d) not defined
7 6 x
If one of the root of the equation 2 x 2 = 0 is x = -9,
x 3 7
then the other two roots are
(a) (2,6) (b) (3,6) (c) (2,7) (d) (3,7)
2r ? 1 2.3r-1 4.5r-1 n
12. If Dr = a b g , then the value of åDr, is
2n ?1 3 n ?1 5n ?1 r = 1
(a) 0 (b) a b g (c) a +b + g (d) a . 2 n + b . 3n + g . 4n
If each element of a 3 ´ 3 matrix is multiplied by 3, then the determinant of the newly formed matrix is
(a) 3 det A (b) 9 det A (c) 27 det A (d) (det A)3
Let a ij denote the element of the ith row and jth column in a 3 ´ 3 determinant
(1 ? i ? 3, 1 ? j ? 3) and let aij = - a ji for every i and j. Then the determinant has all the principal diagonal elements as
(a) 1 (b) ?1 (c) 0 (d) none of these
1 1 1
15. The value of the determinant m C 1 m + 1C1 m + 2 C 1 is equal to
m C 2 m + 1 C2 m + 2 C 2
(a) 1 (b) ?1 (c) 0 (d) none of these
pa qb rc
16. If p + q + r = a + b + c = 0. Then the value of qc ra pb is
rb pc qa
(a) 0 (b)ap + bq + cr (c) 1 (d) none of these
cosa -sina 0
17. Let F (a) = sina cosa 0 then, F ( a) F (a') is equal to
0 0 1
(a) F (a a') (b) F (a / a') (c) F (a + a') (d) F (a -a')
x b b
18. D1 = a x b and D2 = x b are the given determinants, then
a a x a x
(a) D1 = 3 ( D2)2 (b) d (D1) = 3 D2 (c) d (D1) = 3 (D2)2 (d )D1 = 3 D23/2
dx dx
The number of solutions of 2x + y = 4, x ? 2y = 2, 3x + 5y = 6 is
(a) 0 (b) 1 (c) 2 (d) infinitely many.
If the system of equations ax + y + z = 0, x + b y + z = 0 and x + y + c z = 0
(a, b, c ? 1 ) has a non-trivial solution, then the value of 1 + 1 + 1 is
1 ? a 1 ? b 1-c
(a) ?1 (b) 0 (c) 1 (d) none of these
Choose the correct answer
a. Every scalar matrix is an identity matrix
b. Every identity matrix is a scalar matrix
c. Every diagonal matrix is an identity matrix
d. A square matrix whose each element is 1 is an identity matrix.
1 x x+1
22. If f (x) = 2x x (x-1) (x+1)x , then f(100) =
3x(x-1) x(x-1)(x-2) (x+1)x(x-1)
(a) 0 (b) 1 (c) 100 (d) -100
If A and B are square matrices of order 3 such that ½A½ = -1, ½B½ = 3, then ½3AB½ =
(a) ?9 (b) ?81 (c) ?27 (d) 81
x + 1 w w2
24 If w is cube root of unity, then D= w x + w2 1 =
w2 1 x + w
(a) x3 + 1 (b) x3 + w (c) x3 + w2 (d) x3
25. If the system of the equation x +2y ? 3z = 2, (K + 3)z = 3, (2K + 1)y + z = 2
is inconsistent, then K is
(a) ?3 (b) ?1/2 (c) 1 (d) 2
26. If the system of the equation x ? Ky ? z = 0, Kx ? y ? z = 0, x + y ? z = 0 has a
non-zero solution, then the possible value of K are:
(a) ?1,2 (b) 1,2 (c)0,1 (d) ?1,1
27. The parameter on which the value of the determinant
1 a a2
cos (p - d)x cos px cos (p + d)x does not depends upon is
sin ( p - d)x sin px sin (p + d)x
(a) a (b) p (c) d (d) x
x p + y x y
28. The determinant y p + z y z = 0 if
0 x p +y y p +z
x,y,z are in A.P. (b) x,y,z are in G.P.
(c) x,y,z are in H.P. (d) xy, yz, zx are in A.P
Let A be a matrix of order 3 and let D denote the value of determinant A.
Then det (-2A)
(a) ?8 D (b) ?2 D (c) 2 D (d) 8D
If A and B are two square matrices such that B = -A-1 BA, then (A+B)2 =
(a) 0 (b) A2 + B2 (c) A2 +2AB + B2 (d) A+B
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