Which of the following statement is/are correct?
(A) A square matrix \(A = [a_{ij}]\) is called a symmetric matrix if \(a_{ij}=a_{ji}\) for all \(i,j\)
(B) \(A = [a_{ij}]_{m\times m}\) is a diagonal matrix if \(a_{ij} =0\) when \(i = j\)
(C) A square matrix \(A = [a_{ij}]\) is called a skew symmetric matrix, if \(a_{ij} = -a_{ji}\) for all \(i,j\)
(D) The multiplication of diagonal matrices of same order is commutative
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
The definitions for a square matrix \(A = [a_{ij}]\):
Symmetric: \(A' = A\), that is, \(a_{ij} = a_{ji}\) for all i, j
Skew-symmetric: \(A' = -A\), that is, \(a_{ij} = -a_{ji}\) for all i, j
Diagonal: \(a_{ij} = 0\) whenever \(i \ne j\)
For two diagonal matrices of the same order, the product is diagonal and its entries are the products of the matching diagonal entries:
\(\mathrm{diag}(d_1, \ldots, d_n)\cdot\mathrm{diag}(e_1, \ldots, e_n) = \mathrm{diag}(d_1e_1, \ldots, d_ne_n)\)
So a diagonal matrix needs zeros off the diagonal (\(i \ne j\)), and diagonal matrices of the same order commute, since \(d_ie_i = e_id_i\).
Checking the statements
(A) A square matrix \(A = [a_{ij}]\) is called a symmetric matrix if \(a_{ij}=a_{ji}\) for all \(i,j\) — Correct. This is the definition of a symmetric matrix.
(B) \(A = [a_{ij}]_{m\times m}\) is a diagonal matrix if \(a_{ij} =0\) when \(i = j\) — Incorrect. The condition is \(a_{ij} = 0\) when \(i \ne j\), not \(i = j\).
(C) A square matrix \(A = [a_{ij}]\) is called a skew symmetric matrix, if \(a_{ij} = -a_{ji}\) for all \(i,j\) — Correct. This is the definition of a skew-symmetric matrix.
(D) The multiplication of diagonal matrices of same order is commutative — Correct. Both products have diagonal entries \(d_ie_i = e_id_i\).