Question

A = [ 3-2 prove that A^2-A+2I=0
4 -2]

Answer 1

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered}$

Now, we have to prove that,

$\rm :\longmapsto\: {A}^{2} - A + 2I = 0$

So,

Let consider

$\rm :\longmapsto\: {A}^{2}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered} \times \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}9 - 8&-6 + 4\\ 12- 8& - 8 + 4\end{array}\right]\end{gathered}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}1&-2\\ 4& - 4\end{array}\right]\end{gathered}$

Now,

$\rm :\longmapsto\:2I$

$\rm \: = \: 2\begin{gathered}\sf\left[\begin{array}{cc}1&0\\ 0& 1\end{array}\right]\end{gathered}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}2&0\\ 0& 2\end{array}\right]\end{gathered}$

Thus, Now consider,

$\rm :\longmapsto\: {A}^{2} - A + 2I$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}1&-2\\ 4& - 4\end{array}\right]\end{gathered} - \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered} + \begin{gathered}\sf\left[\begin{array}{cc}2&0\\ 0& 2\end{array}\right]\end{gathered}$

can be re-arranged as

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}1&-2\\ 4& - 4\end{array}\right]\end{gathered} + \begin{gathered}\sf\left[\begin{array}{cc}2&0\\ 0& 2\end{array}\right]\end{gathered} - \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered} - \begin{gathered}\sf\left[\begin{array}{cc}3&-2\\ 4& - 2\end{array}\right]\end{gathered}$

$\rm \: = \: \begin{gathered}\sf\left[\begin{array}{cc}0&0\\ 0& 0\end{array}\right]\end{gathered}$

Hence,

$\bf :\longmapsto\: {A}^{2} - A + 2I = 0$

Additional Information :-

1. Matrix multiplication is possible only when number of columns of pre - multiplier is equal to number of rows of post multiplier otherwise matrix multiplication is not defined.

2. Matrix multiplication may or may not be Commutative.

3. Matrix multiplication is Distributive.