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Home > Matrix & Vector calculators > Cofactor of a matrix example
11. Matrix Cofactors example ( Enter your problem )
( Enter your problem )
  1. Definition and Examples
  2. Example-2
 		Other related methods
  1. Addition of two matrix
  2. Multiplication of two matrix
  3. Division of two matrix
  4. Power of a matrix
  5. Transpose of a matrix
  6. Determinant of a matrix
  7. Adjoint of a matrix
  8. Inverse of a matrix
  9. Prove that any two matrix expression is equal or not
  10. Minor of a matrix
  11. Cofactor of a matrix
  12. Trace of a matrix
1. Definition and Examples
(Previous example)		12. Trace of a matrix
(Next method)

2. Example-2


1. Find `"Cofactor"(A)` ...
`A=[[1,2],[4,5]]`

Solution:
`COFACTOR(A)` = 
`COFACTOR`
`1``2`
`4``5`


Cofactor of `1=A_(11) = `
 + 
 `5` 
`=+(5)``=5`


Cofactor of `2=A_(12) = `
 - 
 `4` 
`=-(4)``=-4`


Cofactor of `4=A_(21) = `
 - 
 `2` 
`=-(2)``=-2`


Cofactor of `5=A_(22) = `
 + 
 `1` 
`=+(1)``=1`


The Cofactor matrix of A is `[A_(ij)]`=
`A_(11)``A_(12)`
`A_(21)``A_(22)`
=
`5``-4`
`-2``1`


Method-2 : all Cofactors in matrix form

=
 + 
 `5` 
 - 
 `4` 
 - 
 `2` 
 + 
 `1` 


=
`+(5)``-(4)`
`-(2)``+(1)`


=
````
````


=
`5``-4`
`-2``1`
2. Find `"Cofactor"(A)` ...
`A=[[1,2,3],[4,5,6],[7,8,9]]`

Solution:
`COFACTOR(A)` = 
`COFACTOR`
`1``2``3`
`4``5``6`
`7``8``9`


Cofactor of `1=A_(11) = `
 + 
 `5`  `6` 
 `8`  `9` 
`=+(5 × 9 - 6 × 8)``=+(45 -48)``=-3`


Cofactor of `2=A_(12) = `
 - 
 `4`  `6` 
 `7`  `9` 
`=-(4 × 9 - 6 × 7)``=-(36 -42)``=6`


Cofactor of `3=A_(13) = `
 + 
 `4`  `5` 
 `7`  `8` 
`=+(4 × 8 - 5 × 7)``=+(32 -35)``=-3`


Cofactor of `4=A_(21) = `
 - 
 `2`  `3` 
 `8`  `9` 
`=-(2 × 9 - 3 × 8)``=-(18 -24)``=6`


Cofactor of `5=A_(22) = `
 + 
 `1`  `3` 
 `7`  `9` 
`=+(1 × 9 - 3 × 7)``=+(9 -21)``=-12`


Cofactor of `6=A_(23) = `
 - 
 `1`  `2` 
 `7`  `8` 
`=-(1 × 8 - 2 × 7)``=-(8 -14)``=6`


Cofactor of `7=A_(31) = `
 + 
 `2`  `3` 
 `5`  `6` 
`=+(2 × 6 - 3 × 5)``=+(12 -15)``=-3`


Cofactor of `8=A_(32) = `
 - 
 `1`  `3` 
 `4`  `6` 
`=-(1 × 6 - 3 × 4)``=-(6 -12)``=6`


Cofactor of `9=A_(33) = `
 + 
 `1`  `2` 
 `4`  `5` 
`=+(1 × 5 - 2 × 4)``=+(5 -8)``=-3`


The Cofactor matrix of A is `[A_(ij)]`=
`A_(11)``A_(12)``A_(13)`
`A_(21)``A_(22)``A_(23)`
`A_(31)``A_(32)``A_(33)`
=
`-3``6``-3`
`6``-12``6`
`-3``6``-3`


Method-2 : all Cofactors in matrix form

=
 + 
 `5`  `6` 
 `8`  `9` 
 - 
 `4`  `6` 
 `7`  `9` 
 + 
 `4`  `5` 
 `7`  `8` 
 - 
 `2`  `3` 
 `8`  `9` 
 + 
 `1`  `3` 
 `7`  `9` 
 - 
 `1`  `2` 
 `7`  `8` 
 + 
 `2`  `3` 
 `5`  `6` 
 - 
 `1`  `3` 
 `4`  `6` 
 + 
 `1`  `2` 
 `4`  `5` 


=
`+(5 × 9 - 6 × 8)``-(4 × 9 - 6 × 7)``+(4 × 8 - 5 × 7)`
`-(2 × 9 - 3 × 8)``+(1 × 9 - 3 × 7)``-(1 × 8 - 2 × 7)`
`+(2 × 6 - 3 × 5)``-(1 × 6 - 3 × 4)``+(1 × 5 - 2 × 4)`


=
`+(45 -48)``-(36 -42)``+(32 -35)`
`-(18 -24)``+(9 -21)``-(8 -14)`
`+(12 -15)``-(6 -12)``+(5 -8)`


=
`-3``6``-3`
`6``-12``6`
`-3``6``-3`


This material is intended as a summary. Use your textbook for detail explanation.
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1. Definition and Examples
(Previous example)		12. Trace of a matrix
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