Operations on matrix
case1- Cube of a matrix
Let given matrix be \[\begin{bmatrix}2&1\\0&5\end{bmatrix}\]
then \[A^3\]=AAA That means first multiply A with itself and then the product is mutiplied by A .
Therefore \[A^3\]=\[\begin{bmatrix}2&1\\0&5\end{bmatrix}\]\[\begin{bmatrix}2&1\\0&5\end{bmatrix}\]\[\begin{bmatrix}2&1\\0&5\end{bmatrix}\]
=\[ \begin{bmatrix}4&7\\0&25\end{bmatrix}\]\[\begin{bmatrix}2&1\\0&5\end{bmatrix}\]
=\[ \begin{bmatrix}8&39\\0&125\end{bmatrix}\]
Case2- 5th power of a diagonal matrix
let a diagonal matrix be \[ \begin{bmatrix}2&0\\0&5\end{bmatrix}\]
then \[B^5\] is obtained by taking 5th power of each diagonal element and rest elements are zero.
Hence \[B^5\]=\[ \begin{bmatrix}2^5&0\\0&5^5\end{bmatrix}\]=\[ \begin{bmatrix}32&0\\0&3125\end{bmatrix}\]
Case3-Inverse of a matrix
Given Matrix C=\[ \begin{bmatrix}a&b\\c&d\end{bmatrix}\]
inverse is given by adj C divided by determinant of C
\[ \frac{\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}{ad-bc}\]
Condition of inverse to exist - Matrix must be non singular ...i.e det (C) must not be 0
