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Home > Matrix & Vector calculators > Prove that any two matrix expression is equal or not example
9. Prove that any two matrix expression is equal or not example ( Enter your problem )
( Enter your problem )
  1. Example-1
  2. Examples-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. Example-1
(Previous example)		10. Minor of a matrix
(Next method)

2. Examples-2


Find `A × Adj(A) = |A| × I` ...
`A=[[2,1,1],[-1,2,1],[1,1,1]]`

Solution:
To find LHS = `A × Adj(A)`

`Adj(A)` = 
Adj
`2``1``1`
`-1``2``1`
`1``1``1`


 = 
 + 
 `2`  `1` 
 `1`  `1` 
 - 
 `-1`  `1` 
 `1`  `1` 
 + 
 `-1`  `2` 
 `1`  `1` 
 - 
 `1`  `1` 
 `1`  `1` 
 + 
 `2`  `1` 
 `1`  `1` 
 - 
 `2`  `1` 
 `1`  `1` 
 + 
 `1`  `1` 
 `2`  `1` 
 - 
 `2`  `1` 
 `-1`  `1` 
 + 
 `2`  `1` 
 `-1`  `2` 
T


 = 
`+(2 × 1 - 1 × 1)``-(-1 × 1 - 1 × 1)``+(-1 × 1 - 2 × 1)`
`-(1 × 1 - 1 × 1)``+(2 × 1 - 1 × 1)``-(2 × 1 - 1 × 1)`
`+(1 × 1 - 1 × 2)``-(2 × 1 - 1 × (-1))``+(2 × 2 - 1 × (-1))`
T


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


 = 
`1``2``-3`
`0``1``-1`
`-1``-3``5`
T


 = 
`1``0``-1`
`2``1``-3`
`-3``-1``5`


`A×Adj(A)`=
`2``1``1`
`-1``2``1`
`1``1``1`
×
`1``0``-1`
`2``1``-3`
`-3``-1``5`


=
`2×1+1×2+1×-3``2×0+1×1+1×-1``2×-1+1×-3+1×5`
`-1×1+2×2+1×-3``-1×0+2×1+1×-1``-1×-1+2×-3+1×5`
`1×1+1×2+1×-3``1×0+1×1+1×-1``1×-1+1×-3+1×5`


=
`2+2-3``0+1-1``-2-3+5`
`-1+4-3``0+2-1``1-6+5`
`1+2-3``0+1-1``-1-3+5`


=
`1``0``0`
`0``1``0`
`0``0``1`


`:.``A × Adj(A)` = 
`1``0``0`
`0``1``0`
`0``0``1`
` ->(1)`


To find RHS = `|A| × I`

`|A|` = 
 `2`  `1`  `1` 
 `-1`  `2`  `1` 
 `1`  `1`  `1` 


 =
 `2` × 
 `2`  `1` 
 `1`  `1` 
 `-1` × 
 `-1`  `1` 
 `1`  `1` 
 `+1` × 
 `-1`  `2` 
 `1`  `1` 


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

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

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

`= 2 +2 -3`

`=1`


`|A| × I` = `1` × 
`1``0``0`
`0``1``0`
`0``0``1`
 = 
`1``0``0`
`0``1``0`
`0``0``1`


`:.``|A| × I` = 
`1``0``0`
`0``1``0`
`0``0``1`
` ->(2)`


From (1) and (2)
`:. A × Adj(A)=|A| × I` (proved)

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example-1
(Previous example)		10. Minor of a matrix
(Next method)


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