Inverse matrix Definition and Examples

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8. Inverse matrix example ( Enter your problem )

( Enter your problem )

Definition and Examples
Example-2

Other related methods

7. Adjoint of a matrix
(Previous method)

2. Example-2
(Next example)

1. Definition and Examples

Inverse of a matrix
If `A` is a non singular square matrix then its inverse is
`A^-1 = ("Adj"A)/|A|`

Examples
1. Find `A^-1` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`

Solution:

`|A|`

`3`	`1`	`1`
`-1`	`2`	`1`
`1`	`1`	`1`

`3`

	`2`	`1`
	`1`	`1`

`-1`

	`-1`	`1`
	`1`	`1`

`+1`

	`-1`	`2`
	`1`	`1`

`=3 xx (2 × 1 - 1 × 1) -1 xx (-1 × 1 - 1 × 1) +1 xx (-1 × 1 - 2 × 1)`

`=3 xx (2 -1) -1 xx (-1 -1) +1 xx (-1 -2)`

`=3 xx (1) - -1 xx (-2) +1 xx (-3)`

`= 3 +2 -3`

`=2`

`Adj(A)`

Adj

`3`	`1`	`1`
`-1`	`2`	`1`
`1`	`1`	`1`

	`2`	`1`
	`1`	`1`

	`-1`	`1`
	`1`	`1`

	`-1`	`2`
	`1`	`1`

	`1`	`1`
	`1`	`1`

	`3`	`1`
	`1`	`1`

	`3`	`1`
	`1`	`1`

	`1`	`1`
	`2`	`1`

	`3`	`1`
	`-1`	`1`

	`3`	`1`
	`-1`	`2`

`+(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))`

`+(2 -1)`	`-(-1 -1)`	`+(-1 -2)`
`-(1 -1)`	`+(3 -1)`	`-(3 -1)`
`+(1 -2)`	`-(3 +1)`	`+(6 +1)`

`1`	`2`	`-3`
`0`	`2`	`-2`
`-1`	`-4`	`7`

`1`	`0`	`-1`
`2`	`2`	`-4`
`-3`	`-2`	`7`

`"Now, "A^(-1)=1/|A| × Adj(A)`

`1/(2)` ×

`1`	`0`	`-1`
`2`	`2`	`-4`
`-3`	`-2`	`7`

`1/2`	`0`	`-1/2`
`1`	`1`	`-2`
`-3/2`	`-1`	`7/2`

2. Find `B^-1` ...
`B=[[2,3,1],[0,5,6],[1,1,2]]`

Solution:

`|B|`

`2`	`3`	`1`
`0`	`5`	`6`
`1`	`1`	`2`

`2`

	`5`	`6`
	`1`	`2`

`-3`

	`0`	`6`
	`1`	`2`

`+1`

	`0`	`5`
	`1`	`1`

`=2 xx (5 × 2 - 6 × 1) -3 xx (0 × 2 - 6 × 1) +1 xx (0 × 1 - 5 × 1)`

`=2 xx (10 -6) -3 xx (0 -6) +1 xx (0 -5)`

`=2 xx (4) - -3 xx (-6) +1 xx (-5)`

`= 8 +18 -5`

`=21`

`Adj(B)`

Adj

`2`	`3`	`1`
`0`	`5`	`6`
`1`	`1`	`2`

	`5`	`6`
	`1`	`2`

	`0`	`6`
	`1`	`2`

	`0`	`5`
	`1`	`1`

	`3`	`1`
	`1`	`2`

	`2`	`1`
	`1`	`2`

	`2`	`3`
	`1`	`1`

	`3`	`1`
	`5`	`6`

	`2`	`1`
	`0`	`6`

	`2`	`3`
	`0`	`5`

`+(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)`

`+(10 -6)`	`-(0 -6)`	`+(0 -5)`
`-(6 -1)`	`+(4 -1)`	`-(2 -3)`
`+(18 -5)`	`-(12 +0)`	`+(10 +0)`

`4`	`6`	`-5`
`-5`	`3`	`1`
`13`	`-12`	`10`

`4`	`-5`	`13`
`6`	`3`	`-12`
`-5`	`1`	`10`

`"Now, "B^(-1)=1/|B| × Adj(B)`

`1/(21)` ×

`4`	`-5`	`13`
`6`	`3`	`-12`
`-5`	`1`	`10`

`4/21`	`-5/21`	`13/21`
`2/7`	`1/7`	`-4/7`
`-5/21`	`1/21`	`10/21`

This material is intended as a summary. Use your textbook for detail explanation.
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