Q.If $A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{bmatrix}$, then compute $(A+B)$ and $(B-C)$. Also, verify that $A+(B-C) = (A+B)-C$.

Concept: Matrix Addition — matrices of the same order are added (or subtracted) by adding (or subtracting) corresponding entries.

Step 1: Compute $A+B$

Add entrywise:

Step 2: Compute $B-C$

Subtract entrywise:

Step 3: Verify $A+(B-C) = (A+B)-C$

Compute left side:

Compute right side: …

Matrix addition (and subtraction) works exactly like adding or subtracting numbers, but applied to each corresponding position. If two matrices have the same dimensions, you simply combine the entries that sit in the same row and column. This is why the operation is called element-wise.

The property we're verifying — $A+(B-C) = (A+B)-C$ — is the associative law for addition and subtraction. It says that grouping doesn't matter: subtracting $C$ from $B$ first, then adding $A$, gives the same result as adding $A$ and $B$ first, then subtracting $C$. This works because matrix addition is commutative and associative, just like ordinary addition.

Let's compute step by step.

1. Compute $A+B$ Add each corresponding entry:

2. Compute $B-C$ Subtract each entry of $C$ from the corresponding entry of $B$:

3. Compute $A + (B-C)$ Add $A$ to the result from step 2:

4. Compute $(A+B)-C$ Subtract $C$ from the result of step 1: …

Method: Computing Combinations and Verifying an Addition Identity

Use this when asked to compute sums/differences and then verify an algebraic identity like $A+(B-C)=(A+B)-C$. Compute each side independently and compare.

Steps

Step 1: Compute the required combinations entrywise.

Addition and subtraction are position by position on matrices of equal order: $(A+B)_{ij}=a_{ij}+b_{ij}$, $(B-C)_{ij}=b_{ij}-c_{ij}$. Mind subtraction of negatives.

Step 2: Evaluate both sides of the identity separately. …

Common Mistakes

Mistake 1: Reversing the order of a difference.

Why it's wrong: $B-C$ is not $C-B$; subtraction is order-sensitive. Correct approach: subtract in the exact order written.

Mistake 2: Sign errors when a subtracted entry is negative.

Why it's wrong: $0-(-2)=2$, not $-2$; dropped signs break the verification. Correct approach: rewrite $b_{ij}-c_{ij}$ carefully when $c_{ij}<0$. …