For \(a, b \gt 0\), if \(P = \begin{bmatrix}0 & -a\\2a & b\end{bmatrix}\) and \(Q = \begin{bmatrix}b & a\\-b & 0\end{bmatrix}\) are two matrices such that \(PQ = \begin{bmatrix}2 & 0\\3 & 8\end{bmatrix}\), then the value of \((a + b)^{ab}\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
Multiplying the matrices row by column:
\( \displaystyle PQ = \begin{bmatrix}0\cdot b + (-a)(-b) & 0\cdot a + (-a)\cdot 0\\2a\cdot b + b(-b) & 2a\cdot a + b\cdot 0\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}ab & 0\\2ab - b^2 & 2a^2\end{bmatrix} \)
Comparing with \(\begin{bmatrix}2 & 0\\3 & 8\end{bmatrix}\):
\(2a^2 = 8,\quad a = 2 \ (a \gt 0)\)
\(ab = 2,\quad b = 1\)
These also satisfy the third entry: \(2ab - b^2 = 4 - 1 = 3\). Hence
\((a + b)^{ab} = 3^2 = 9\)