Which of the following are TRUE?
(A) The inverse of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \displaystyle\frac{1}{27}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \).
(B) The determinant of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is 29.
(C) The inverse of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \displaystyle\frac{1}{29}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \)
(D) The adjoint of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (D) only
Let \( \mathrm{M}=\begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \).
\( \displaystyle |\mathrm{M}|=(3)(7)-(-3)(2) \)
\( \displaystyle =21+6=27 \)
For a 2 × 2 matrix, the adjoint is found by swapping the diagonal entries and changing the signs of the other two:
\( \displaystyle \mathrm{adj}\,\mathrm{M}=\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \)
\( \displaystyle \mathrm{M}^{-1}=\frac{1}{|\mathrm{M}|}\,\mathrm{adj}\,\mathrm{M}=\frac{1}{27}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \)
So the determinant is 27, \( \mathrm{adj}\,\mathrm{M}=\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \) and \( \mathrm{M}^{-1}=\frac{1}{27}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \).
Checking the statements
(A) The inverse of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \frac{1}{27}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \). — Correct. This is exactly \( \mathrm{M}^{-1} \).
(B) The determinant of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is 29. — Incorrect. The determinant is 21 + 6 = 27, not 29.
(C) The inverse of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \frac{1}{29}\begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \) — Incorrect. It divides by 29 instead of the determinant 27.
(D) The adjoint of \( \begin{bmatrix}3 & -3\\2 & 7\end{bmatrix} \) is \( \begin{bmatrix}7 & 3\\-2 & 3\end{bmatrix} \) — Correct. Diagonal entries swapped, the other two with signs changed.