Which of the following statements are correct?
(A) \(A = [a_{ij}]_{n\times n}\) is a diagonal matrix if \(a_{ij}=0\) when \(i = j\)
(B) A square matrix \(A = [a_{ij}]\) is called a symmetric matrix if \(a_{ij}=a_{ji}\) for all \(i,j\)
(C) A square matrix \(A = [a_{ij}]\) is called a skew-symmetric matrix if \(a_{ij}=-a_{ji}\) for all \(i,j\)
(D) For every square matrix A, there exist an identity matrix of the same order such that IA = AI= I
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
In a diagonal matrix every entry off the main diagonal is zero: \(a_{ij} = 0\) whenever \(i \ne j\).
A square matrix is symmetric if \(A^T = A\), that is \(a_{ij} = a_{ji}\) for all i, j, and skew-symmetric if \(A^T = -A\), that is \(a_{ij} = -a_{ji}\) for all i, j.
The identity matrix I of the same order as A leaves A unchanged: \(IA = AI = A\).
So a diagonal matrix needs \(a_{ij} = 0\) for \(i \ne j\), and \(IA = AI = A\).
Checking the statements
(A) \(A = [a_{ij}]_{n\times n}\) is a diagonal matrix if \(a_{ij}=0\) when \(i = j\) — Incorrect. The zero entries must be those with \(i \ne j\); \(a_{ii} = 0\) only makes the diagonal zero.
(B) 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.
(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) For every square matrix A, there exist an identity matrix of the same order such that IA = AI= I — Incorrect. \(IA = AI = A\), not I.