Q.If $A$ and $B$ are square matrices of the same order, then

Concept: Matrix Transpose — the transpose of a product reverses the order, and scalar multiplication commutes with transposition.

1. By the reversal law for transposes: $(AB)' = B' A'$.
2. A scalar factor is unaffected by transposition: $(kA)' = k A'$.
3. First factor the scalar: $[k(A - B)]' = k (A - B)'$. …

Why this works — the core idea

Matrix transpose is like flipping a matrix over its main diagonal: rows become columns and columns become rows. The key property for products is that when you transpose $AB$, you must reverse the multiplication order. Why? Because the $(i,j)$ entry of $AB$ comes from row $i$ of $A$ and column $j$ of $B$. After transposing, that entry moves to position $(j,i)$, which now comes from row $j$ of $B'$ and column $i$ of $A'$ — hence $B'A'$.

For scalar multiplication, each entry is simply multiplied by $k$, so transposing just carries that $k$ along.

Let's apply these ideas step by step.

1. For $(AB)'$ The transpose of a product of two matrices equals the product of their transposes in reverse order.

This is a standard result — you can verify it by checking the $(i,j)$ entry on both sides.

2. For $(kA)'$ If every entry of $A$ is multiplied by $k$, then transposing gives a matrix where every entry is still $k$ times the corresponding entry of $A'$. So

The scalar simply factors out.

3. For $[k(A - B)]'$ Work from the inside out. First, $A - B$ is just entry-wise subtraction. Then multiply by $k$. Then transpose.
   • Inside the brackets: $k(A - B)$ means each entry of $(A - B)$ is scaled by $k$. …

Method: Simplify transposes with the transpose rules

Use this for questions asking for the transpose of products, scalar multiples, and combinations.

Steps

Step 1: Recall the three working rules.

Step 2: Handle products with the reversal law.

For a product, transpose each factor and reverse their order — a very common slip is to keep the original order. …

Common Mistakes

Mistake 1: Writing $(AB)' = A'B'$ instead of $B'A'$.

Why it's wrong: the transpose of a product reverses the factor order. Correct approach: always flip to $B'A'$.

Mistake 2: Reversing the order for a sum or difference. …