Let A = [aᵢⱼ]ₙₓₙ and B = [bᵢⱼ]ₙₓₙ. Then which of the following is/are true?

(A) AB = BA

(B) (AB)⁻¹ = B⁻¹ A⁻¹

(C) $(AB)^T = B^T A^T$

(D) AB = 0 ⇒ A = 0 or B = 0

Choose the correct answer from the options given below:

Let $A = [a_{ij}]_{n \times n}$ and $B = [b_{ij}]_{n \times n}$. Checking each statement:

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Statement (A): $AB = BA$

Matrix multiplication is generally not commutative.

Consider:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

$AB = \begin{bmatrix} 1 \times 5 + 2 \times 7 & 1 \times 6 + 2 \times 8 \\ 3 \times 5 + 4 \times 7 & 3 \times 6 + 4 \times 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

$BA = \begin{bmatrix} 5 \times 1 + 6 \times 3 & 5 \times 2 + 6 \times 4 \\ 7 \times 1 + 8 \times 3 & 7 \times 2 + 8 \times 4 \end{bmatrix} = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$

Since $\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} \neq \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$, the statement is FALSE.

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Statement (B): $(AB)^{-1} = B^{-1}A^{-1}$

This is a standard property of matrix inverses.

$(AB) \times (B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = A(I)A^{-1} = AA^{-1} = I$

This holds when both $A$ and $B$ are invertible matrices.

The statement is TRUE.

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Statement (C): $(AB)^T = B^T A^T$

This is the transpose property for matrix products. When transposing a product, the order of matrices reverses.

The statement is TRUE.

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Statement (D): $AB = 0 \Rightarrow A = 0$ or $B = 0$

Two non-zero matrices can multiply to give the zero matrix.

Consider:

$A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$

$AB = \begin{bmatrix} 1 \times 0 + 0 \times 1 & 1 \times 0 + 0 \times 0 \\ 0 \times 0 + 0 \times 1 & 0 \times 0 + 0 \times 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Even though neither $A$ nor $B$ is zero, their product is the zero matrix.

The statement is FALSE.

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Only statements (B) and (C) are true.