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

1. Definition and Examples


Minor of a matrix `(2 xx 2)`
`|A|=|[a,b],[c,d]|`
minor of `a = d`
minor of `b = c`
minor of `c = b`
minor of `d = a`
Minor of a matrix `(3 xx 3)`
`|A|=|[a,b,c],[d,e,f],[g,h,i]|`
minor of `a = |[e,f],[h,k]|`
minor of `b = |[d,f],[g,k]|`
minor of `c = |[d,e],[g,h]|`
minor of `d = |[b,c],[h,i]|`
minor of `e = |[a,c],[g,i]|`
minor of `f = |[a,b],[g,h]|`
minor of `g = |[b,c],[e,f]|`
minor of `h = |[a,c],[d,f]|`
minor of `i = |[a,b],[d,e]|`
`A=[[1,2,3],[4,5,6],[7,8,9]]`
minor of `1 = A_11 = |[5,6],[8,9]| = 45-48 = -3`
minor of `2 = A_12 = |[4,6],[7,9]| = 36-42 = -6`
minor of `3 = A_13 = |[4,5],[7,8]| = 32-35 = -3`

minor of `4 = A_21 = |[2,3],[8,9]| = 18-24 = -6`
minor of `5 = A_22 = |[1,3],[7,9]| = 9-21 = -12`
minor of `6 = A_23 = |[1,2],[7,8]| = 8-14 = -6`

minor of `7 = A_31 = |[2,3],[5,6]| = 12-15 = -3`
minor of `8 = A_32 = |[1,3],[4,6]| = 6-12 = -6`
minor of `9 = A_33 = |[1,2],[4,5]| = 5-8 = -3`

The minor 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]]`
Examples
1. Find `"Minor"(A)` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`

Solution:
`MINOR(A)` = 
`MINOR`
`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`


2. Find `"Minor"(B)` ...
`B=[[2,3,1],[0,5,6],[1,1,2]]`

Solution:
`MINOR(B)` = 
`MINOR`
`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`




This material is intended as a summary. Use your textbook for detail explanation.
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9. Prove that any two matrix expression is equal or not
(Previous method)		2. Example-2
(Next example)


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