Question

If A2 = I, where I is the identity matrix, which matrix correctly represents matrix A?

Answer 1

We know that $\boldsymbol{A^2}=\boldsymbol{A\times A}$

We also know that $\boldsymbol{I}=\begin{bmatrix} 1 &0 \\ 0&1 \end{bmatrix}$

Now, it has been given to us that $\boldsymbol{A^2}=\boldsymbol{I}$

Therefore, we will have to find the correct $\boldsymbol{A}$ from the given options and we find that when:

$\boldsymbol{A}=\begin{bmatrix} 3 &-2 \\ 4&-3 \end{bmatrix}$

then $\boldsymbol{A^2}=\boldsymbol{A\times A}=\begin{bmatrix} 3 &-2 \\ 4&-3 \end{bmatrix}\times \begin{bmatrix} 3 &-2 \\ 4&-3 \end{bmatrix}=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

Therefore, from the above given options, Option C is the correct option.

Therefore, $\boldsymbol{A}=\begin{bmatrix} 3 & -2\\ 4 & -3 \end{bmatrix}$ is the correct answer.