Let \(A = (a_{ij})\) and \(B = (b_{ij})\) are square matrices of same order.
(A) The number of possible matrices of order \(2 \times 2\) with entries −1, 0, 1 is 81.
(B) A + A' is skew symmetric matrix
(C) \(A \cdot A^{-1} = 0\), \(|A| \ne 0\)
(D) A is skew symmetric matrix if \(a_{ij} = -a_{ji}\) for all i, j
(E) (AB)' = A'B'
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
A \(2 \times 2\) matrix has 4 entries, and each can be chosen in 3 ways:
\(3^4 = 81\)
The standard results needed:
\((A + A')' = A' + A\)
\(A \cdot A^{-1} = I\)
\((AB)' = B'A'\)
So there are 81 such matrices, A + A′ is symmetric, \(A \cdot A^{-1} = I\), and \((AB)' = B'A'\).
Checking the statements
(A) The number of possible matrices of order \(2 \times 2\) with entries −1, 0, 1 is 81. — Correct. 3 choices for each of 4 entries: \(3^4 = 81\).
(B) A + A' is skew symmetric matrix — Incorrect. \((A + A')' = A + A'\), so it is symmetric.
(C) \(A \cdot A^{-1} = 0\), \(|A| \ne 0\) — Incorrect. \(A \cdot A^{-1}\) is the identity matrix I, not 0.
(D) A is skew symmetric matrix if \(a_{ij} = -a_{ji}\) for all i, j — Correct. This is the definition of a skew symmetric matrix, A′ = −A.
(E) (AB)' = A'B' — Incorrect. The order reverses: \((AB)' = B'A'\).