Let \(A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}\). Which of the following statements are correct?
(A) \(A + A' = \begin{bmatrix} 4 & 2 \\ 2 & 8 \end{bmatrix}\)
(B) \(A - A'\) is a skew-symmetric matrix whose element in the first row and second column is \(-4\).
(C) \(|A'| = 11\)
(D) \(|A - A'| = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
The transpose of A is
\( \displaystyle A' = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix} \)
Adding and subtracting element by element:
\( \displaystyle A + A' = \begin{bmatrix} 4 & 2 \\ 2 & 8 \end{bmatrix} \)
\( \displaystyle A - A' = \begin{bmatrix} 0 & -4 \\ 4 & 0 \end{bmatrix} \)
The determinants:
\(|A'| = |A| = 2 \times 4 - (-1) \times 3 = 11\)
\(|A - A'| = 0 \times 0 - (-4) \times 4 = 16\)
So \(A + A'\) has rows (4, 2) and (2, 8), \(A - A'\) has rows (0, −4) and (4, 0), \(|A'| = 11\) and \(|A - A'| = 16\).
Checking the statements
(A) \(A + A' = \begin{bmatrix} 4 & 2 \\ 2 & 8 \end{bmatrix}\) — Correct. Adding A and A' element by element gives this matrix.
(B) \(A - A'\) is a skew-symmetric matrix whose element in the first row and second column is \(-4\). — Correct. \(A - A'\) has rows (0, −4) and (4, 0), and its transpose is its negative.
(C) \(|A'| = 11\) — Correct. \(|A'| = |A| = 8 + 3 = 11\).
(D) \(|A - A'| = 0\) — Incorrect. \(|A - A'| = 0 + 16 = 16\).