Let P be a matrix such that \( \mathrm{P}=\begin{bmatrix}1 & -1\\0 & 3\end{bmatrix} \), then which of the following statements are True?
A. P is a symmetric matrix
B. P is not a skew-symmetric matrix
C. The determinant of P is non-zero
D. P is an invertible matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B, C and D only
\( \displaystyle \mathrm{P}^T=\begin{bmatrix}1 & 0\\-1 & 3\end{bmatrix} \)
\( \displaystyle |\mathrm{P}|=(1)(3)-(-1)(0) \)
\( \displaystyle =3 \)
So \( \mathrm{P}^T\ne\mathrm{P} \), the diagonal entries of P are 1 and 3 (not zero), and \( |\mathrm{P}|=3 \).
Checking the statements
(A) P is a symmetric matrix — Incorrect. The (1, 2) entry is −1 but the (2, 1) entry is 0.
(B) P is not a skew-symmetric matrix — Correct. A skew-symmetric matrix has a zero diagonal, but P has 1 and 3.
(C) The determinant of P is non-zero — Correct. \( |\mathrm{P}|=3\ne 0 \).
(D) P is an invertible matrix — Correct. \( |\mathrm{P}|=3\ne 0 \), so P is invertible.