Matrices and Determinants
Invertible Matrices
IIT-JEE 1995
Grade 12
Question:
If $A$ is a square matrix such that $A^2 = O$ but $A \neq O$, then:
(1) $|A| = 0$
(2) $|A| = 1$
(3) $A$ is invertible
(4) $A = I$
Step-by-Step Solution
Key Concept: Suppose, for contradiction, that $A$ were invertible. Multiply both sides of $A^2 = O$ by $A^{-1}$ and see what contradiction arises with the assumption $A \neq O$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)