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

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

Step 1: Find $A$ from $A'$. Since $A'$ is $3 \times 2$, $A$ is $2 \times 3$:

Step 2: Compute $A+B$ and $A-B$:

Step 3: Transpose these results:

Step 4: Compute $A'+B'$ and $A'-B'$:

Both pairs match, verifying the properties.

The core idea here is that transposition is a linear operation — it respects addition and subtraction. When you flip rows to columns, the order of addition doesn't matter. So $(A+B)' = A' + B'$ is not just a coincidence; it's a fundamental property that makes matrix algebra consistent.

We are given $A'$ (which is the transpose of $A$) and $B$. To verify the identities, we first need to find $A$ from $A'$, then compute $A+B$ and $A-B$, take their transposes, and compare with $A' + B'$ and $A' - B'$.

Let's proceed step by step.

1. Find $A$ from $A'$ Since $A'$ is the transpose of $A$, we have $A = (A')'$. Given $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$, which is a $3 \times 2$ matrix, its transpose $A$ will be a $2 \times 3$ matrix:

2. Write down $B$

   $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, which is also $2 \times 3$. Good — $A$ and $B$ have the same dimensions, so addition and subtraction are defined.

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

4. Transpose these results

5. Now compute $A' + B'$ and $A' - B'$ First, find $B'$:

Then:

6. Compare We see that $(A+B)'$ exactly matches $A' + B'$, and $(A-B)'$ exactly matches $A' - B'$. Both identities are verified.

Method: Verifying a transpose identity when $A'$ is given instead of $A$

When the problem supplies $A'$ rather than $A$, first recover $A=(A')'$, then verify the required identity by computing both sides.

Steps

Step 1: Recover $A$

Since transposing twice returns the original, $A=(A')'$ — turn the given $A'$'s rows into columns. Confirm $A$ and $B$ then share an order so they can be added.

Step 2: Compute the left side

Form $A\pm B$ entry-wise and transpose it.

Step 3: Compute the right side and compare

Form $A'\pm B'$ (you already have $A'$; transpose $B$ for $B'$) and check it matches the left side.

Common Mistakes

Mistake 1: Adding the given $A'$ directly to $B$

Why it's wrong: $A'$ is $3\times2$ while $B$ is $2\times3$, so they cannot be added; you must first recover $A=(A')'$. Correct approach: transpose the given $A'$ back to $A$ before combining.

Mistake 2: Forgetting that $(A')'=A$

Why it's wrong: skipping this step means working with the wrong matrix throughout. Correct approach: reconstruct $A$ from $A'$ at the very start.