Question

Let A be an n $\times$ n real matrix such that $A^2=I$ and Y be an n-dimensional vector. Then the linear system of equations Ax = Y has

• A. no solution
• B. unique solution
• C. more than one but infinitely many dependent solutions
• D. infinitely many dependent solutions

Answer 1

The correct option is B unique solution

$\Rightarrow AA=I\Rightarrow A=A^{-1}$
$\Rightarrow$A is invertible i.e. $A^{-1}$ exists.
$\Rightarrow$ A is non-singular

Hence for the system $A_{n\times n}{X}_{n\times 1}=Y_{n\times 1}$ unique solution exist.