A matrix P \(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\) is such that P\(^T\) = - P, where P\(^T\) is the transpose of P. If b = 1, then P is
A
\(\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\)
B
\(\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\)
C
\(\begin{pmatrix}0 & 1\\ -1 & 1\end{pmatrix}\)
D
\(\begin{pmatrix}1 & -1\\ -1 & 0\end{pmatrix}\)
Verified Correct Answer: A.
Step-by-Step Mathematical Solution
Given P\(^T\) = -P, b = 1
If P = \(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\), then, P\(^T\) = \(\begin{pmatrix}a & c\\ b & d\end{pmatrix}\)
\(\begin{pmatrix}a & c\\ b & d\end{pmatrix}\) = -\(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\) = \(\begin{pmatrix}-a & -1\\ -c & -d\end{pmatrix}\)
comparing
a = -a ⇒ 0, b = 1, c = -1, d = -d ⇒ 0
\(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\) = \(\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #69896)