Question

If the square of the matrix $\binom{a\quad b}{a -a}$ is the unit matrix, then $b$ is equal to

• A. $\frac{a }{1+a^{2}}$
• B. $\frac{1-a^{2}}{a}$
• C. $\frac{1+a^{2}}{a}$
• D. $\frac{a}{1-a^{2}}$
• E. $1 + a^2$

Answer 1

Answer: (B)

Given, $\begin{bmatrix}a & b \\ a & -a\end{bmatrix}^{2}=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix} a & b \\ a & -a\end{bmatrix} \begin{bmatrix} a & b \\ a & -a\end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} $
$\Rightarrow \begin{bmatrix} a^{2}+a b & a b-a b \\ a^{2}-a^{2} & a b+a^{2}\end{bmatrix} =\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}^{2}+a b & 0 \\ 0 & a b+a^{2}\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
On equating the elements, we get
$a^{2}+a b=1$
$\Rightarrow a b=1-a^{2} \Rightarrow b=\frac{1-a^{2}}{a}$