Which of the following statements are correct?
(A) The matrix \(\begin{bmatrix}6&-3\\-4&2\end{bmatrix}\) has no inverse.
(B) If \(A = \begin{bmatrix}x&0\\1&1\end{bmatrix}\) and \(A^2 = \begin{bmatrix}9&0\\4&1\end{bmatrix}\), then \(x = 3\) or \(x = -3\).
(C) The inverse of \(\begin{bmatrix}3&2\\4&3\end{bmatrix}\) is \(\begin{bmatrix}3&-2\\-4&3\end{bmatrix}\).
(D) If A is a \(3 \times 2\) matrix and B is a \(2 \times 4\) matrix, then BA is defined.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
A square matrix has an inverse only when its determinant is not zero. For the first matrix
\(6(2) - (-3)(-4) = 12 - 12 = 0\)
For the matrix in (B):
\( \displaystyle A^2 = \begin{bmatrix}x^2&0\\x + 1&1\end{bmatrix} \)
Comparing with the given \(A^2\), both conditions must hold:
\(x^2 = 9\)
\(x + 1 = 4\)
The second gives x = 3, and x = 3 also satisfies the first. For (C), a matrix is the inverse when the product is I:
\( \displaystyle \begin{bmatrix}3&2\\4&3\end{bmatrix}\begin{bmatrix}3&-2\\-4&3\end{bmatrix} = \begin{bmatrix}9 - 8&-6 + 6\\12 - 12&-8 + 9\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} \)
A product BA is defined only when the number of columns of B equals the number of rows of A.
So the first matrix has determinant 0, x = 3 only, the product in (C) is I, and B has 4 columns while A has 3 rows.
Checking the statements
(A) The matrix \(\begin{bmatrix}6&-3\\-4&2\end{bmatrix}\) has no inverse. — Correct. Its determinant is 0.
(B) If \(A = \begin{bmatrix}x&0\\1&1\end{bmatrix}\) and \(A^2 = \begin{bmatrix}9&0\\4&1\end{bmatrix}\), then \(x = 3\) or \(x = -3\). — Incorrect. x + 1 = 4 rules out x = −3; only x = 3 works.
(C) The inverse of \(\begin{bmatrix}3&2\\4&3\end{bmatrix}\) is \(\begin{bmatrix}3&-2\\-4&3\end{bmatrix}\). — Correct. Their product is I.
(D) If A is a \(3 \times 2\) matrix and B is a \(2 \times 4\) matrix, then BA is defined. — Incorrect. B has 4 columns but A has 3 rows.