Which of the following statements are TRUE?A. If the product of two m…

The correct answer is Option (3) → A and D only

A product of two non-zero matrices can be the zero matrix:

    \( \displaystyle \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}=\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix} \)

Matrix multiplication is associative, (AB)C = A(BC), but not commutative, so the order of the factors cannot be changed.

In a skew symmetric matrix \( a_{ji}=-a_{ij} \). For i = j this gives

    \( \displaystyle a_{ii}=-a_{ii} \)

    \( \displaystyle a_{ii}=0 \)

Two diagonal matrices multiply entry by entry along the diagonal:

    \( \displaystyle \begin{bmatrix}a_1 & 0\\0 & a_2\end{bmatrix}\begin{bmatrix}b_1 & 0\\0 & b_2\end{bmatrix}=\begin{bmatrix}a_1b_1 & 0\\0 & a_2b_2\end{bmatrix} \)

So a zero product does not need a zero factor, (AB)C = A(BC) but not A(CB) in general, a skew symmetric matrix has a zero diagonal, and diagonal matrices of the same order commute.

Checking the statements

(A) If the product of two matrices is a zero matrix, it is not necessary that one of the matrix is a zero matrix. — Correct. \( \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix} \) is the zero matrix, though neither factor is.

(B) For any three matrices A, B and C, (AB)C = A(CB). — Incorrect. The associative law gives A(BC); CB may differ from BC or may not even be defined.

(C) Diagonal elements of a skew symmetric matrix can be non-zero. — Incorrect. \( a_{ii}=-a_{ii} \) forces every diagonal element to be 0.

(D) Multiplication of diagonal matrices of same order is always commutative. — Correct. The diagonal entries of the product are \( a_ib_i=b_ia_i \).