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

Concept: Matrix Addition — add corresponding entries after scaling.

Step 1: Multiply matrix $A$ by the scalar $2$:

Step 2: Subtract matrix $B$ entry-wise:

$2A - B = \begin{bmatrix} 2-3 & 4-(-1) & 6-3 \\ 4-(-1) & 6-0 & 2-2 \end{bmatrix}$ …

Why this works

Matrix addition and subtraction are the simplest operations in linear algebra — they happen entry by entry. If two matrices have the same dimensions (here both are $2 \times 3$), you can add or subtract them by working on each position independently. Scaling a matrix (multiplying by a constant like $2$) means multiplying every single entry by that constant.

So $2A - B$ means: first scale $A$ by $2$, then subtract $B$ from the result, element-wise.

Step-by-step

1. Write down $A$ and $B$ clearly.

Both are $2 \times 3$ — so the operation is defined.

2. Compute $2A$ — multiply every entry of $A$ by $2$.

3. Subtract $B$ from $2A$ — subtract corresponding entries.

For the first row:

• First column: $2 - 3 = -1$
• Second column: $4 - (-1) = 4 + 1 = 5$
• Third column: $6 - 3 = 3$

For the second row:

• First column: $4 - (-1) = 4 + 1 = 5$
• Second column: $6 - 0 = 6$ …

Method: Computing a linear combination $2A - B$

Use this for expressions combining scalar multiples of matrices, such as $2A - B$.

Steps

Step 1: Confirm the orders match.

All matrices in the combination must have the same order for addition/subtraction to be defined.

Step 2: Apply each scalar entry-wise. …

Common Mistakes

Mistake 1: Sign errors subtracting negative entries.

Why it's wrong: $4 - (-1) = 5$, not $3$; a dropped double-negative corrupts the result. Correct approach: subtracting a negative is adding its magnitude.

Mistake 2: Applying the scalar to only some entries.

Why it's wrong: $2A$ doubles every entry of $A$, not just the first row/column. Correct approach: multiply all entries by the scalar. …