Let \(A = \begin{bmatrix}2&3\\-1&4\end{bmatrix}\). Which of the following statements are correct?
(A) \(A + A' = \begin{bmatrix}4&2\\2&8\end{bmatrix}\)
(B) \(A - A' = \begin{bmatrix}0&4\\-4&0\end{bmatrix}\)
(C) \(AA' = A'A\)
(D) \(A'A\) is a symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
The transpose of A turns its rows into columns:
\( \displaystyle A' = \begin{bmatrix}2&-1\\3&4\end{bmatrix} \)
\( \displaystyle A + A' = \begin{bmatrix}4&2\\2&8\end{bmatrix} \)
\( \displaystyle A - A' = \begin{bmatrix}0&4\\-4&0\end{bmatrix} \)
Multiplying row by column:
\( \displaystyle AA' = \begin{bmatrix}4 + 9&-2 + 12\\-2 + 12&1 + 16\end{bmatrix} = \begin{bmatrix}13&10\\10&17\end{bmatrix} \)
\( \displaystyle A'A = \begin{bmatrix}4 + 1&6 - 4\\6 - 4&9 + 16\end{bmatrix} = \begin{bmatrix}5&2\\2&25\end{bmatrix} \)
So \(A + A' = \begin{bmatrix}4&2\\2&8\end{bmatrix}\), \(A - A' = \begin{bmatrix}0&4\\-4&0\end{bmatrix}\), \(AA' = \begin{bmatrix}13&10\\10&17\end{bmatrix}\) and \(A'A = \begin{bmatrix}5&2\\2&25\end{bmatrix}\).
Checking the statements
(A) \(A + A' = \begin{bmatrix}4&2\\2&8\end{bmatrix}\) — Correct. The off-diagonal elements are \(3 + (-1) = 2\).
(B) \(A - A' = \begin{bmatrix}0&4\\-4&0\end{bmatrix}\) — Correct. The first-row, second-column element is \(3 - (-1) = 4\).
(C) \(AA' = A'A\) — Incorrect. The first elements are 13 and 5.
(D) \(A'A\) is a symmetric matrix. — Correct. Both off-diagonal elements of \(A'A\) are 2.