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Pre-Calculus Matrices

Matrix operations (add, subtract, scalar multiply)

20 practice questions 0 video lessons Theory + worked examples

Matrix Operations

California Pre-Calculus • Standard N-VM.8 • Matrices

Matrix Operations is the opening topic of Matrices in the California Common Core State Standards. It is aligned to Standard N-VM.8, which requires students to add, subtract, and scalar-multiply matrices.

Matrix operations add and subtract matrices of the same size entry by entry, and multiply every entry by a scalar.

California Pre-Calculus › Matrices › Matrix Operations  —  Standard N-VM.8

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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.
Adding matrices entrywise Two matrices of the same size add entry by entry. 1 2 3 4 + 5 6 7 8 = 6 8 10 12
Matrices of the same size add entry by entry.
Rules Rules Rules add / subtract: same size, entrywise kA: multiply every entry by k size m × n (rows × columns)
The rules at a glance.

Entrywise operations:

\[(A+B)_{ij}=A_{ij}+B_{ij},\qquad (kA)_{ij}=k\,A_{ij}\]
matrix addition adds corresponding entries; scalar multiplication multiplies every entry
Dimensions: \(A+B\) needs equal sizes; \(kA\) keeps the size of \(A\).

How to combine matrices

  1. Check sizes match before adding or subtracting.
  2. Add/subtract corresponding entries.
  3. Scale by multiplying each entry.
  4. 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.

\[=\begin{bmatrix}1+5&2+6\\3+7&4+8\end{bmatrix}=\begin{bmatrix}6&8\\10&12\end{bmatrix}\]
sum is 6, 8, 10, 12
Example 2 — Subtract matrices
Compute \(\begin{bmatrix}5&0\\2&7\end{bmatrix}-\begin{bmatrix}1&3\\4&2\end{bmatrix}\).
Solution

Subtract entrywise.

\[=\begin{bmatrix}4&-3\\-2&5\end{bmatrix}\]
difference is 4, negative 3, negative 2, 5
Example 3 — Scalar multiply
Compute \(3\begin{bmatrix}2&-1\\0&4\end{bmatrix}\).
Solution

Multiply every entry by 3.

\[=\begin{bmatrix}6&-3\\0&12\end{bmatrix}\]
result is 6, negative 3, 0, 12
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 \(B\).

\[2A=\begin{bmatrix}2&4\\6&0\end{bmatrix},\quad 2A-B=\begin{bmatrix}2&3\\1&-2\end{bmatrix}\]
result is 2, 3, 1, negative 2

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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