Let \(A = \begin{bmatrix}1&0\\0&0\end{bmatrix}\) and \(B = \begin{bmatrix}0&1\\0&0\end{bmatrix}\). Consider the following statements:
(A) \(AB = B\)
(B) \(BA = O\)
(C) \(AB = BA\)
(D) \((A + B)^2 = A^2 + 2AB + B^2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Multiplying row by column:
\( \displaystyle AB = \begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}0&1\\0&0\end{bmatrix} = \begin{bmatrix}0&1\\0&0\end{bmatrix} \)
\( \displaystyle BA = \begin{bmatrix}0&1\\0&0\end{bmatrix}\begin{bmatrix}1&0\\0&0\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix} \)
Also \(A^2 = A\), \(B^2 = O\) and \(A + B = \begin{bmatrix}1&1\\0&0\end{bmatrix}\):
\( \displaystyle (A + B)^2 = \begin{bmatrix}1&1\\0&0\end{bmatrix} \)
\( \displaystyle A^2 + 2AB + B^2 = \begin{bmatrix}1&0\\0&0\end{bmatrix} + \begin{bmatrix}0&2\\0&0\end{bmatrix} = \begin{bmatrix}1&2\\0&0\end{bmatrix} \)
So AB = B, BA = O, \(AB \ne BA\), and \((A + B)^2 = \begin{bmatrix}1&1\\0&0\end{bmatrix}\) while \(A^2 + 2AB + B^2 = \begin{bmatrix}1&2\\0&0\end{bmatrix}\).
Checking the statements
(A) \(AB = B\) — Correct. \(AB = \begin{bmatrix}0&1\\0&0\end{bmatrix} = B\).
(B) \(BA = O\) — Correct. Every element of BA is 0.
(C) \(AB = BA\) — Incorrect. AB has 1 in the first row, second column, but BA = O.
(D) \((A + B)^2 = A^2 + 2AB + B^2\) — Incorrect. The first-row, second-column elements are 1 and 2; the identity needs \(AB = BA\).