1. Definition and Examples
Adjoint of a matrix
Let `A` be a square matrix then Adjoint of matrix A is transpose of Cofactor matrix of A. It is denoted by adj`A`
Adjoint matrix is also called adjugate matrix.
`A=[[1,2,3],[4,5,6],[7,8,9]]`
To find adjoint of a matrix, we have to first find Cofactor matrix of A and then of transpose of this Cofactor matrix A.
Cofactor of `1 = A_11 = +|[5,6],[8,9]| = +(45-48) = -3`
Cofactor of `2 = A_12 = -|[4,6],[7,9]| = -(36-42) = 6`
Cofactor of `3 = A_13 = +|[4,5],[7,8]| = +(32-35) = -3`
Cofactor of `4 = A_21 = -|[2,3],[8,9]| = -(18-24) = 6`
Cofactor of `5 = A_22 = +|[1,3],[7,9]| = +(9-21) = -12`
Cofactor of `6 = A_23 = -|[1,2],[7,8]| = -(8-14) = 6`
Cofactor of `7 = A_31 = +|[2,3],[5,6]| = +(12-15) = -3`
Cofactor of `8 = A_32 = -|[1,3],[4,6]| = -(6-12) = 6`
Cofactor of `9 = A_33 = +|[1,2],[4,5]| = +(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]]`
Now find transpose of `A_ij`
Adj`A=[[-3,6,-3],[6,-12,6],[-3,6,-3]]^T`
`=[[-3,6,-3],[6,-12,6],[-3,6,-3]]^T`
Examples
1. Find `Adj(A)` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`
Solution:
| `Adj(A)` | = | | Adj | | `3` | `1` | `1` | | | `-1` | `2` | `1` | | | `1` | `1` | `1` | |
|
|
| = | | `+(2 × 1 - 1 × 1)` | `-(-1 × 1 - 1 × 1)` | `+(-1 × 1 - 2 × 1)` | | | `-(1 × 1 - 1 × 1)` | `+(3 × 1 - 1 × 1)` | `-(3 × 1 - 1 × 1)` | | | `+(1 × 1 - 1 × 2)` | `-(3 × 1 - 1 × (-1))` | `+(3 × 2 - 1 × (-1))` | |
| T |
|
| = | | `+(2 -1)` | `-(-1 -1)` | `+(-1 -2)` | | | `-(1 -1)` | `+(3 -1)` | `-(3 -1)` | | | `+(1 -2)` | `-(3 +1)` | `+(6 +1)` | |
| T |
|
| = | | `1` | `2` | `-3` | | | `0` | `2` | `-2` | | | `-1` | `-4` | `7` | |
| T |
|
| = | | `1` | `0` | `-1` | | | `2` | `2` | `-4` | | | `-3` | `-2` | `7` | |
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2. Find `Adj(A)` ...
`A=[[2,3,1],[0,5,6],[1,1,2]]`
Solution:
| `Adj(A)` | = | | Adj | | `2` | `3` | `1` | | | `0` | `5` | `6` | | | `1` | `1` | `2` | |
|
|
| = | | `+(5 × 2 - 6 × 1)` | `-(0 × 2 - 6 × 1)` | `+(0 × 1 - 5 × 1)` | | | `-(3 × 2 - 1 × 1)` | `+(2 × 2 - 1 × 1)` | `-(2 × 1 - 3 × 1)` | | | `+(3 × 6 - 1 × 5)` | `-(2 × 6 - 1 × 0)` | `+(2 × 5 - 3 × 0)` | |
| T |
|
| = | | `+(10 -6)` | `-(0 -6)` | `+(0 -5)` | | | `-(6 -1)` | `+(4 -1)` | `-(2 -3)` | | | `+(18 -5)` | `-(12 +0)` | `+(10 +0)` | |
| T |
|
| = | | `4` | `6` | `-5` | | | `-5` | `3` | `1` | | | `13` | `-12` | `10` | |
| T |
|
| = | | `4` | `-5` | `13` | | | `6` | `3` | `-12` | | | `-5` | `1` | `10` | |
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This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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