Let A, B, C be square matrices of equal orders and I be the identity matrix of same order, then which of the following are TRUE?
(A) (AB)C = A(BC)
(B) AB = BA
(C) A(B + C) = AB + AC
(D) AI = IA = A
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
For square matrices of the same order, matrix multiplication is associative, is distributive over addition, and has I as its identity. It is not commutative in general. For example, with
\( \displaystyle A=\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix} \)
\( \displaystyle AB=\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix} \)
\( \displaystyle BA=\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix} \)
So (AB)C = A(BC), A(B + C) = AB + AC and AI = IA = A always hold, while AB and BA can differ.
Checking the statements
(A) (AB)C = A(BC) — Correct. Matrix multiplication is associative.
(B) AB = BA — Incorrect. In the example, AB ≠ O but BA = O.
(C) A(B + C) = AB + AC — Correct. Multiplication is distributive over addition.
(D) AI = IA = A — Correct. I is the multiplicative identity.
NCERT (2026-27 edition):
Properties of Multiplication of Matrices
