Question

If the square of the matrix $\begin{pmatrix} a & b \\ a & -a \end{pmatrix}$ 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. \( 1+a^2 \)

Hint:

The square of a matrix \( \begin{pmatrix} a & b c & -a \end{pmatrix} \) is always a scalar matrix of the form \( (a^2+bc)I \). Setting this equal to \( I \) gives \( a^2+bc=1 \). In our case, \( c=a \), so \( a^2+ab=1 \).

Answer 1

The Correct Option is B