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Home > Matrix & Vector calculators > Multiplication of two matrix example
2. Matrix Multiplication 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
1. Addition of two matrix
(Previous method)		2. Example-2
(Next example)

1. Definition and Examples


1. Scalar Multiplication

If `A=[[2,3,1],[0,5,6],[1,1,2]]` then
`2*A=2*[[2,3,1],[0,5,6],[1,1,2]]=[[2*2,2*3,2*1],[2*0,2*5,2*6],[2*1,2*1,2*2]]=[[4,6,2],[0,10,12],[2,2,4]]`
2. Multiplication of matrices

If `A=[[a_11,a_12,a_13],[a_21,a_22,a_23],[a_31,a_32,a_33]]` and `B=[[b_11,b_12,b_13],[b_21,b_22,b_23],[b_31,b_32,b_33]]` then

`A xx B = [[a_11,a_12,a_13],[a_21,a_22,a_23],[a_31,a_32,a_33]] xx [[b_11,b_12,b_13],[b_21,b_22,b_23],[b_31,b_32,b_33]]`

`= [[a_11 b_11 + a_12 b_21 + a_13 b_31, a_11 b_12 + a_12 b_22 + a_13 b_32, a_11 b_13 + a_12 b_23 + a_13 b_33], [a_21 b_11 + a_22 b_21 + a_23 b_31, a_21 b_12 + a_22 b_22 + a_23 b_32, a_21 b_13 + a_22 b_23 + a_23 b_33], [a_31 b_11 + a_32 b_21 + a_33 b_31, a_31 b_12 + a_32 b_22 + a_33 b_32, a_31 b_13 + a_32 b_23 + a_33 b_33] ]`
3. Properties of Matrix Multiplication

1. Matrix Multiplication is not Commutative
`AB != BA`

2. Matrix Multiplication is Associative
`A(BC) = (AB)C`

3. Matrix Multiplication is Distributive
`A(B+C) = AB+AC`

4. Matrix Multiplication by Unit matrix
`AI=IA=A`

5. `A * A^-1 = A^-1 * A = I`
Examples
1. Find `A × B` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`,`B=[[5,0,-2],[7,-6,0],[1,1,2]]`

Solution:
`A×B`=
`3``1``1`
`-1``2``1`
`1``1``1`
×
`5``0``-2`
`7``-6``0`
`1``1``2`


=
`3×5+1×7+1×1``3×0+1×-6+1×1``3×-2+1×0+1×2`
`-1×5+2×7+1×1``-1×0+2×-6+1×1``-1×-2+2×0+1×2`
`1×5+1×7+1×1``1×0+1×-6+1×1``1×-2+1×0+1×2`


=
`15+7+1``0-6+1``-6+0+2`
`-5+14+1``0-12+1``2+0+2`
`5+7+1``0-6+1``-2+0+2`


=
`23``-5``-4`
`10``-11``4`
`13``-5``0`


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

Solution:
`A×B`=
`2``1``1`
`-1``2``1`
`1``1``1`
×
`2``3``1`
`0``5``6`
`1``1``2`


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


=
`4+0+1``6+5+1``2+6+2`
`-2+0+1``-3+10+1``-1+12+2`
`2+0+1``3+5+1``1+6+2`


=
`5``12``10`
`-1``8``13`
`3``9``9`




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1. Addition of two matrix
(Previous method)		2. Example-2
(Next example)


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