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Home > Matrix & Vector calculators > Inverse of a matrix example
8. Inverse matrix 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
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` 
T


 = 
`+(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`


`"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` 
T


 = 
`+(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`


`"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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7. Adjoint of a matrix
(Previous method)		2. Example-2
(Next example)


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