If A = \( \begin{bmatrix}0 & 3\\2 & 0\end{bmatrix} \) and A-1 = K (adj A), then the value of K is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( -\displaystyle\frac{1}{6} \)
For an invertible matrix, \( \mathrm{A}^{-1}=\frac{1}{|\mathrm{A}|}\,\mathrm{adj\,A} \), so \( K=\frac{1}{|\mathrm{A}|} \), the reciprocal of the determinant.
\( \displaystyle |\mathrm{A}|=\begin{vmatrix}0 & 3\\2 & 0\end{vmatrix} \)
\( \displaystyle =0\times 0-3\times 2=-6 \)
\( \displaystyle K=\frac{1}{|\mathrm{A}|}=-\frac{1}{6} \)
Why the other options are incorrect
Option (2) 6 — Incorrect. 6 is the size of |A| without its sign; K is the reciprocal of |A|, not |A| itself.
Option (3) \( \displaystyle\frac{1}{6} \) — Incorrect. This drops the minus sign: \( |\mathrm{A}|=0-6=-6 \), not 6.
Option (4) -6 — Incorrect. −6 is |A| itself; K is its reciprocal.