Which of the following statement 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:

Let's check each statement one by one

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Statement (A): $A = [a_{ij}]_{n \times n}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$

Incorrect

A diagonal matrix is one where all the off-diagonal elements are zero, i.e., $a_{ij} = 0$ when $i \neq j$.

Statement (A) says $a_{ij} = 0$ when $i = j$, which means the diagonal elements themselves are zero. That's the opposite of the definition!

For example, $\begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$ is a diagonal matrix — the non-zero elements are on the diagonal ($i = j$), and zeros are off the diagonal ($i \neq j$).

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Statement (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 exact definition of a symmetric matrix. It means $A = A^T$.

For example, $\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ — here $a_{12} = a_{21} = 2$. ✔️

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Statement (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 exact definition of a skew-symmetric matrix. It means $A = -A^T$.

When $i = j$:

$a_{ii} = -a_{ii}$

$2a_{ii} = 0$

$a_{ii} = 0$

So all diagonal elements of a skew-symmetric matrix must be zero.

For example, $\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$ ✔️

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Statement (D): For every square matrix $A$, there exists an identity matrix of the same order such that $IA = AI = I$

Incorrect

The correct property of the identity matrix is:

$IA = AI = A$ (not $I$)

The identity matrix leaves $A$ unchanged when multiplied. The statement claims $IA = AI = I$, which would mean every square matrix equals $I$ — clearly wrong.

For example, let $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$IA = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} = A \neq I$

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Statements $(B)$ and $(C)$ are correct.