If \( A=\begin{bmatrix} a & 0\\ 0 & 0\end{bmatrix} \) and \( B=\begin{bmatrix} 0 & 0\\ 0 & b\end{bmatrix} \) then the matrix \( A^{3}B^{3} \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A null matrix
Powers of a diagonal matrix are found by raising each diagonal entry to that power:
\( \displaystyle A^{3}=\begin{bmatrix} a^{3} & 0\\ 0 & 0\end{bmatrix} \)
\( \displaystyle B^{3}=\begin{bmatrix} 0 & 0\\ 0 & b^{3}\end{bmatrix} \)
\( \displaystyle A^{3}B^{3}=\begin{bmatrix} a^{3}\cdot 0+0\cdot 0 & a^{3}\cdot 0+0\cdot b^{3}\\ 0\cdot 0+0\cdot 0 & 0\cdot 0+0\cdot b^{3}\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix} 0 & 0\\ 0 & 0\end{bmatrix} \)
So \( A^{3}B^{3} \) is the null matrix.
Shortcut: AB = O already (each non-zero entry of A meets a zero of B), so \( A^{3}B^{3}=A^{2}(AB)B^{2}=O \).
Why the other options are incorrect
Option (2) An identity matrix — Incorrect. The product has no non-zero entry at all, so it cannot be the identity matrix.
Option (3) \( \begin{bmatrix} a^{3} & 0\\ 0 & b^{3}\end{bmatrix} \) — Incorrect. This matrix is \( A^{3}+B^{3} \), not \( A^{3}B^{3} \).
Option (4) \( \begin{bmatrix} a^{3}b^{3} & 0\\ 0 & 0\end{bmatrix} \) — Incorrect. For diagonal matrices the product multiplies the entries in the same position: \( a^{3}\times 0=0 \) in the (1, 1) place, not \( a^{3}b^{3} \).