Let \(A = \begin{bmatrix}1&2\\2&4\end{bmatrix}\) and \(B = \begin{bmatrix}2&-2\\-1&1\end{bmatrix}\). Consider the following statements:
(A) \(AB = O\), the zero matrix of order 2.
(B) \(BA = O\)
(C) \((A + B)^2 = A^2 + BA + B^2\)
(D) Neither A nor B is invertible.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
Multiplying rows by columns:
\( \displaystyle AB = \begin{bmatrix}2 - 2&-2 + 2\\4 - 4&-4 + 4\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix} \)
\( \displaystyle BA = \begin{bmatrix}2 - 4&4 - 8\\-1 + 2&-2 + 4\end{bmatrix} = \begin{bmatrix}-2&-4\\1&2\end{bmatrix} \)
Expanding \((A + B)^2\) keeping the order of each product:
\((A + B)^2 = A^2 + AB + BA + B^2\)
\((A + B)^2 = A^2 + BA + B^2\), since \(AB = O\)
So \(AB = O\) with both A and B non-zero, \(BA \ne O\), and \((A + B)^2 = A^2 + BA + B^2\).
Checking the statements
(A) \(AB = O\), the zero matrix of order 2. — Correct. Every element of AB works out to 0.
(B) \(BA = O\) — Incorrect. \(BA = \begin{bmatrix}-2&-4\\1&2\end{bmatrix}\), so \(AB = O\) does not force \(BA = O\).
(C) \((A + B)^2 = A^2 + BA + B^2\) — Correct. The term AB in \(A^2 + AB + BA + B^2\) is O.
(D) Neither A nor B is invertible. — Correct. If A had an inverse, \(B = A^{-1}(AB) = A^{-1}O = O\); if B had one, \(A = (AB)B^{-1} = O\); but neither is O.