Q58 / 150 · MHT CET 2025 (21 Apr Shift 2) · Mathematics

Let $A$ be a non-singular matrix of order $n$ and $|A|=k$, then $(\operatorname{adj} A)^{-1}$ is

  • Correct answer
    A
    $\frac{\mathrm{A}}{\mathrm{k}}$
  • B
    $\mathrm{k}^{\mathrm{n}-1}(\operatorname{adj} \mathrm{~A})$
  • C
    $\mathrm{k}^{\mathrm{n}-2} \mathrm{~A}$
  • D
    kA

Solution

A non-singular matrix $A$ of order $n$ with $|A|=k$ satisfies the fundamental relationship $\operatorname{adj} A = |A| A^{-1} = k A^{-1}$.

The inverse of $\operatorname{adj} A$ is then given by $(\operatorname{adj} A)^{-1} = (k A^{-1})^{-1} = \frac{1}{k} (A^{-1})^{-1} = \frac{1}{k} A$.

This result corresponds to option A, $\frac{A}{k}$.

Hence, the correct answer is option: 'A'.

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