Q.If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that

Concept: Matrix Transpose — the transpose of a sum (or difference) equals the sum (or difference) of the transposes.

Step 1: Compute $A + B$ and $A - B$.

Step 2: Transpose each result.

Step 3: Compute $A'$ and $B'$ separately, then add/subtract.

Both pairs match exactly, verifying the properties.

Why This Works: The Intuition

Matrix transpose is a simple operation — it flips rows into columns and columns into rows. When you add two matrices, you add corresponding entries. So if you transpose after adding, you're just flipping the positions of those sums. But if you transpose each matrix first and then add, you're adding the flipped entries — which is exactly the same thing, because addition is commutative and works entry-by-entry. The same logic holds for subtraction. This property is called linearity of the transpose: it respects addition and scalar multiplication.

Let's verify it concretely with the given $3 \times 3$ matrices.

Step-by-Step Verification

1. Compute $A + B$ and $A - B$

First, add and subtract the matrices entry-wise.

2. Transpose $A + B$ and $A - B$

Transpose means: row $i$ becomes column $i$.

3. Compute $A'$ and $B'$ individually

Transpose $A$ and $B$:

4. Add and subtract the transposes

Now add $A'$ and $B'$:

This matches $(A + B)'$ exactly.

Now subtract $B'$ from $A'$:

This matches $(A - B)'$ exactly.

Method: Verifying that the transpose distributes over addition/subtraction

To verify $(A\pm B)' = A'\pm B'$, compute both sides independently and check they are identical — this confirms the linearity of the transpose for the given matrices.

Steps

Step 1: Compute the left side

Add (or subtract) $A$ and $B$ entry-wise, then transpose the result.

Step 2: Compute the right side

Transpose $A$ and $B$ separately to get $A'$ and $B'$, then add (or subtract) them.

Step 3: Compare entry-by-entry

The two matrices must match in every position; a single mismatch signals an arithmetic error to recheck.

Common Mistakes

Mistake 1: Transposing the two sides inconsistently

Why it's wrong: the point is to compare $(A+B)'$ against $A'+B'$; forgetting to transpose one side, or transposing $A$ but not $B$, breaks the check. Correct approach: fully compute each side separately before comparing.

Mistake 2: Sign slips in the $(A-B)'$ verification

Why it's wrong: subtracting entries such as $3-(-5)=8$ is error-prone; a sign mistake makes the two sides disagree. Correct approach: subtract each corresponding entry carefully, watching negatives.