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:

Statement (A): A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$

This is the standard definition of a symmetric matrix. The element at position (row $i$, column $j$) equals the element at position (row $j$, column $i$).

Example:

$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$

Here $a_{12} = 2 = a_{21}$, $a_{13} = 3 = a_{31}$, $a_{23} = 5 = a_{32}$

Statement (A) is correct.

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

The condition is incorrect. A diagonal matrix has $a_{ij} = 0$ when $i \neq j$ (non-diagonal elements are zero, not the diagonal elements).

The correct definition requires all off-diagonal elements to be zero.

Example of diagonal matrix:

$D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$

Statement (B) is incorrect.

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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 $i, j$

This is the standard definition of a skew-symmetric matrix. Each element is the negative of its transpose position. This also implies all diagonal elements must be zero since $a_{ii} = -a_{ii}$ gives $a_{ii} = 0$.

Example:

$A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$

Here $a_{12} = 2 = -(-2) = -a_{21}$

Statement (C) is correct.

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Statement (D): The multiplication of diagonal matrices of same order is commutative

For diagonal matrices $D_1$ and $D_2$, the product $D_1 \times D_2 = D_2 \times D_1$.

When multiplying diagonal matrices, only corresponding diagonal elements multiply:

$D_1 = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}, D_2 = \begin{bmatrix} c & 0 \\ 0 & d \end{bmatrix}$

$D_1 \times D_2 = \begin{bmatrix} ac & 0 \\ 0 & bd \end{bmatrix} = D_2 \times D_1$

Since multiplication of real numbers is commutative ($ac = ca$ and $bd = db$), the order doesn't matter.

Statement (D) is correct.

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Correct statements: (A), (C), and (D) only