Matrix Addition, Subtraction, and Scalar Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Addition, Subtraction, and Scalar Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Matrix addition and subtraction are performed entry by entry on matrices of the same dimensions. Scalar multiplication multiplies every entry of a matrix by a single number (the scalar).

Adding matrices is like adding two spreadsheets cell by cell. If spreadsheet A has sales for January and B has sales for February, then A + B gives total sales in each cell. Scalar multiplication is like giving everyone in the spreadsheet a 10% raise—multiply every entry by 1.1.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Addition and subtraction require matrices of the same size and work entry by entry. Scalar multiplication scales every entry uniformly.

Common stuck point: You can only add or subtract matrices with identical dimensions. A 2 \times 3 matrix cannot be added to a 3 \times 2 matrix.

Sense of Study hint: Line up the matrices so corresponding entries are directly above or below each other before adding or subtracting.

Worked Examples

Example 1

easy
Compute \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}.

Solution

  1. 1
    Step 1: Add corresponding entries: a_{ij} + b_{ij}.
  2. 2
    Step 2: \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}.
  3. 3
    Check: Each entry is the sum of the entries in the same position ✓

Answer

\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}
Matrix addition adds corresponding entries element by element. Both matrices must have the same dimensions for addition to be defined.

Example 2

medium
Compute 3 \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Compute 2\begin{bmatrix} 4 \\ -3 \end{bmatrix}.

Example 2

medium
If A = \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix} and B = \begin{bmatrix} 2 & 4 \\ 1 & -2 \end{bmatrix}, find A - 2B.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

matrix definition