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Home > Matrix & Vector calculators > Power of a matrix example
4. Matrix Power 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
3. Division of two matrix
(Previous method)		2. Example-2
(Next example)

1. Definition and Examples


1. Power of matrix

`A^2=A*A`
`A^3=A^2*A`
`A^4=A^3*A`
`A^5=A^4*A`
Examples
1. Find `A^2` ...
`A=[[3,1,1],[-1,2,1],[1,1,1]]`

Solution:
`A^2`=`A×A`=
`3``1``1`
`-1``2``1`
`1``1``1`
×
`3``1``1`
`-1``2``1`
`1``1``1`


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


=
`9-1+1``3+2+1``3+1+1`
`-3-2+1``-1+4+1``-1+2+1`
`3-1+1``1+2+1``1+1+1`


=
`9``6``5`
`-4``4``2`
`3``4``3`


`A^2` = 
`3``1``1`
`-1``2``1`
`1``1``1`
2
 = 
`9``6``5`
`-4``4``2`
`3``4``3`


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

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


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


=
`4+0+1``6+15+1``2+18+2`
`0+0+6``0+25+6``0+30+12`
`2+0+2``3+5+2``1+6+4`


=
`5``22``22`
`6``31``42`
`4``10``11`


`B^2` = 
`2``3``1`
`0``5``6`
`1``1``2`
2
 = 
`5``22``22`
`6``31``42`
`4``10``11`




This material is intended as a summary. Use your textbook for detail explanation.
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3. Division of two matrix
(Previous method)		2. Example-2
(Next example)


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