adjoint of a matrix

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3 May 2007 23:31:58 IST

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3 May 2007 23:31:58 IST

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adjoint of a matrix

Mathematics ,

May be many of u would be knowing this.... incase u dnt check this out...

Let A = [a_ij]

then adj (A) = [C_ij] whr C_ij is the cofactor of a_ij

now let A is given by | a₁₁ a₁₂ a₁₃ .......... a_1n |

| a₂₁ a₂₂a₂₃.......... a_2n |

| ..................................|

|...................................|

| a_n1 a_n2a_n3.......... a_nn |

adj (A) = | C₁₁ C₁₂ C₁₃ .......... C_1n|

| C₂₁ C₂₂C₂₃.......... C_2n|

| ................................ ..|

|.................................. .|

| C_n1 C_n2C_n3.......... C_nn |

now we have a property in detrminants which states that "if all the elemnts of a prticular row or coloumn is multiplied with its cofactor then thr algebraic sum will be same as determinant's and if they are multiplied by cofactor of any other's row (or coloumn's ) elemnt, then its value will be 0"

using this property...

we'll be having

adj (A) = | |A| 0 0 .......... 0|

| 0 |A| 0......... 0|

| ...........................|

|............................|

| 0 0 0.........|A| |

here the diagonal elemts will be |A| and rest 0 (by the above property)

thrfore adj (A) = |A| I_n

also thr's this property if all the entries of a determinant of order n is replaced by thr corresponding cofactors then value of new-determinant D is D = |A|^n-1[power cofactor formula]

thrfore using this, here

|adj (A)| = |A|^n-1

this can be genralised to |adj (adj (adj..... (n tymes)..(adj (A))))|= |A|^{[n-1)^n]}

Hopefully It helps.....

Dnt go much in proof... just rember the formula...and the property... that will do

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Comments (5)

karthik

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Joined: 25 Feb 2007 01:17:42 IST

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4 May 2007 00:07:56 IST

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thanx didi..........it was asked a q ........and was answered by ruhi...........

Manasi

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Joined: 1 Dec 2006 11:13:27 IST

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8 May 2007 13:31:21 IST

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no comments. :((

karthik

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Joined: 25 Feb 2007 01:17:42 IST

Posts: 1010

8 May 2007 13:36:49 IST

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http://www.goiit.com/posts/list/algebra-determinants-bhut-phaltu-problem-hai-11377.htm#63709
the link for that.........
dont worry didi.......others will post..........

Vinod vm

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Joined: 27 Dec 2006 16:38:24 IST

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12 May 2007 22:52:17 IST

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nice !! it really helped ..

Rajlaxmi Choudhury

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Joined: 16 Jun 2007 07:41:20 IST

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16 Jun 2007 16:13:08 IST

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fantastic! i used the formula it works

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