Question

Let A be a \(2 \times 2\) real matrix with entries from \(\{0, 1\}\) and \(|A| \ne 0\). Consider the following two statements :

(P) If \(A \ne I_2\), then \(|A| = -1\)
(Q) If \(|A| = 1\), then \(\operatorname{tr}(A) = 2\),

where \(I_2\) denotes \(2 \times 2\) identity matrix and \(\operatorname{tr}(A)\) denotes the sum of the diagonal entries of A. Then :

• A. (P) is true and (Q) is false
• B. Both (P) and (Q) are false
• C. Both (P) and (Q) are true
• D. (P) is false and (Q) is true

Hint:

List all 0-1 matrices of order 2 with non-zero determinant and test each statement on them.

Answer 1

The Correct Option is D

With entries 0 or 1, \(|A| = ad - bc \ne 0\) leaves six matrices:
\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\), \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\), \(\begin{pmatrix}1&0\\1&1\end{pmatrix}\) with \(|A| = 1\), and \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\), \(\begin{pmatrix}0&1\\1&1\end{pmatrix}\), \(\begin{pmatrix}1&1\\1&0\end{pmatrix}\) with \(|A| = -1\).
(P): \(A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\ne I_2\) has \(|A| = 1\ne -1\), so (P) is false.
(Q): every matrix with \(|A|=1\) has trace 2, so (Q) is true.
Answer: (4) (P) is false and (Q) is true