Question

if a=(2,5)(1,3);find the matrix Mif a²-2a+3i=M​

Answer 1

Given matrix,

$\longrightarrow A=\left[\begin{array}{cc}2&5\\1&3\end{array}\right]$

And given the equation,

$\longrightarrow M=A^2-2A+3I$

We've to find matrix M.

Since order of A is 2, so are orders of I and M.

$\longrightarrow I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

Let us find $A^2.$

$\longrightarrow A^2=\left[\begin{array}{cc}2&5\\1&3\end{array}\right]\cdot\left[\begin{array}{cc}2&5\\1&3\end{array}\right]$

$\longrightarrow A^2=\left[\begin{array}{cc}2^2+5\times1&5(2+3)\\1(2+3)&5\times1+3^2\end{array}\right]$

$\longrightarrow A^2=\left[\begin{array}{cc}9&25\\5&14\end{array}\right]$

Then the equation becomes,

$\longrightarrow M=\left[\begin{array}{cc}9&25\\5&14\end{array}\right]-2\left[\begin{array}{cc}2&5\\1&3\end{array}\right]+3\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

$\longrightarrow M=\left[\begin{array}{cc}9&25\\5&14\end{array}\right]-\left[\begin{array}{cc}4&10\\2&6\end{array}\right]+\left[\begin{array}{cc}3&0\\0&3\end{array}\right]$

$\longrightarrow M=\left[\begin{array}{cc}9-4+3&25-10+0\\5-2+0&14-6+3\end{array}\right]$

$\longrightarrow\underline{\underline{M=\left[\begin{array}{cc}8&15\\3&11\end{array}\right]}}$