Question

If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$ and I is the unit matrix of order $2$, then $A^2$ equals

• A. $4A–3I$
• B. $3A-4I$
• C. $A-I$
• D. $A+I$

Answer 1

The correct option is A

$4A–3I$

Calculate the value of ${\mathit{A}}^{\mathbf{2}}$:

Given $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$

$$\begin{gathered} A^2=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right]..........\left(i\right) \end{gathered}$$

Now find $4A–3I$

$$\begin{gathered} 4A–3I=4\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]-3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow 4A–3I=\left[\begin{array}{cc}8& -4\\ -4& 8\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right] \\ \Rightarrow 4A–3I=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right] \\ \Rightarrow 4A–3I=A^2\mathbf{}\mathbf{}\mathbf{}\mathbf{\left(}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{\right(}\mathit{i}\mathbf{\left)}\mathbf{\right)}\mathbf{} \end{gathered}$$

Hence, the correct option is (A).