Which of the following matrices are invertible?
(A) \( \begin{bmatrix}2&2\\3&3\end{bmatrix} \)
(B) \( \begin{bmatrix}2&3\\3&2\end{bmatrix} \)
(C) \( \begin{bmatrix}3&2\\2&3\end{bmatrix} \)
(D) \( \begin{bmatrix}3&3\\3&3\end{bmatrix} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
A square matrix is invertible exactly when its determinant is non-zero. For a 2 × 2 matrix \( \begin{bmatrix}a&b\\c&d\end{bmatrix} \) the determinant is \( ad-bc \):
(A) \( 2\times3-2\times3=0 \)
(B) \( 2\times2-3\times3=-5 \)
(C) \( 3\times3-2\times2=5 \)
(D) \( 3\times3-3\times3=0 \)
So the determinants are 0, −5, 5 and 0.
Checking the statements
(A) \( \begin{bmatrix}2&2\\3&3\end{bmatrix} \) — Incorrect. The determinant is 0, so it is not invertible.
(B) \( \begin{bmatrix}2&3\\3&2\end{bmatrix} \) — Correct. The determinant is −5, which is non-zero.
(C) \( \begin{bmatrix}3&2\\2&3\end{bmatrix} \) — Correct. The determinant is 5, which is non-zero.
(D) \( \begin{bmatrix}3&3\\3&3\end{bmatrix} \) — Incorrect. The determinant is 0, so it is not invertible.