Pre-Calculus
Matrices
Matrix operations (add, subtract, scalar multiply)
20 practice questions
0 video lessons
Theory + worked examples
Theory
A matrix is a rectangular array of numbers, described by its dimensions \(m\times n\) (rows by columns). The basic operations are entrywise:
- Addition / subtraction: only for matrices of the same size; add or subtract corresponding entries.
- Scalar multiplication: multiply every entry by the scalar.
Sizes must match to add. A \(2\times 3\) and a \(3\times 2\) matrix cannot be added.
Matrices of the same size add entry by entry.
The rules at a glance.
Entrywise operations:
\[(A+B)_{ij}=A_{ij}+B_{ij},\qquad (kA)_{ij}=k\,A_{ij}\]
Dimensions: \(A+B\) needs equal sizes; \(kA\) keeps the size of \(A\).
How to combine matrices
- Check sizes match before adding or subtracting.
- Add/subtract corresponding entries.
- Scale by multiplying each entry.
- Order of operations: scalar-multiply before adding, as with numbers.
Example 1 — Add matrices
Add \(\begin{bmatrix}1&2\\3&4\end{bmatrix}+\begin{bmatrix}5&6\\7&8\end{bmatrix}\).
Solution
Add corresponding entries.
\[=\begin{bmatrix}1+5&2+6\\3+7&4+8\end{bmatrix}=\begin{bmatrix}6&8\\10&12\end{bmatrix}\]
Example 2 — Subtract matrices
Compute \(\begin{bmatrix}5&0\\2&7\end{bmatrix}-\begin{bmatrix}1&3\\4&2\end{bmatrix}\).
Solution
Subtract entrywise.
\[=\begin{bmatrix}4&-3\\-2&5\end{bmatrix}\]
Example 3 — Scalar multiply
Compute \(3\begin{bmatrix}2&-1\\0&4\end{bmatrix}\).
Solution
Multiply every entry by 3.
\[=\begin{bmatrix}6&-3\\0&12\end{bmatrix}\]
Example 4 — Combine operations
Compute \(2A-B\) for \(A=\begin{bmatrix}1&2\\3&0\end{bmatrix}\), \(B=\begin{bmatrix}0&1\\5&2\end{bmatrix}\).
Solution
Scale \(A\), then subtract \(B\).
\[2A=\begin{bmatrix}2&4\\6&0\end{bmatrix},\quad 2A-B=\begin{bmatrix}2&3\\1&-2\end{bmatrix}\]
Common pitfalls
Same size to add. Mismatched dimensions can't be combined.
Scalar hits every entry. Don't multiply just the first row or column.
Keep entries in place. Add row-\(i\) column-\(j\) to the matching entry, never a shifted one.
Frequently asked questions
How do you add two matrices?
They must be the same size; then add corresponding entries.
What is scalar multiplication of a matrix?
Multiplying every entry of the matrix by the scalar.
What does the size of a matrix mean?
Its dimensions \(m\times n\): \(m\) rows and \(n\) columns.
Can you add matrices of different sizes?
No. Addition and subtraction require identical dimensions.
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