Which of the following statement are correct?

(A) $A = [a_{ij}]_{n×n}$ 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 $|A = A| = I$

Choose the correct answer from the options given below:

(B) and (C) only

The correct answer is Option (4) → (B) and (C) only

Check each statement:

(A) A = [a_ij] is a diagonal matrix if a_ij = 0 when i = j.

→ Incorrect, because for a diagonal matrix, 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.

→ Correct ✔

(C) A square matrix A = [a_ij] is called a skew-symmetric matrix if a_ij = −a_ji for all i, j.

→ Correct ✔

(D) For every square matrix A, there exist an identity matrix of the same order such that |A = A| = I.

→ Incorrect, because |A| denotes determinant; determinant cannot equal the identity matrix. Also, AA = I only if A is invertible.

Correct statements: (B) and (C)