Matrices

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Represent data and transformations using matrices.
  • Add, subtract, and scale matrices.
  • Multiply matrices (row-by-column).
  • Interpret matrices in systems, transformations, and data contexts.

Key Ideas

A matrix is a rectangular array of numbers.

Example of a \(2 \times 3\) matrix: \[ \begin{bmatrix} 1 & 4 & 2 \\ 3 & 0 & 5 \end{bmatrix} \]

Dimensions: rows × columns

  • Row 1: \([1,\;4,\;2]\)
  • Row 2: \([3,\;0,\;5]\)

Operations:

  • Addition/Subtraction: elementwise, only for same-size matrices
    \[ A + B = (a_{ij} + b_{ij}) \]

  • Scalar Multiplication: multiply each entry
    \[ kA = (ka_{ij}) \]

  • Matrix Multiplication: row-by-column
    \[ (AB)_{ij} = \text{Row}_i(A) \cdot \text{Column}_j(B) \]

Multiplication is not commutative: \[ AB \ne BA \]

Common Problem Types

Adding or Subtracting Matrices

Only possible when dimensions match.

Example:
\[ \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 1\\ 0 & -2 \end{bmatrix} = \begin{bmatrix} 6 & 3\\ 3 & 2 \end{bmatrix} \]

Scalar Multiplication

Multiply each element.

Example:
\[ 2 \begin{bmatrix} 3 & -1\\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 6 & -2\\ 8 & 4 \end{bmatrix} \]

Matrix Multiplication (Row-by-Column)

Multiply rows of first by columns of second.

Example:
\[ \begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} 3\\ 4 \end{bmatrix} = 1(3) + 2(4) = 11 \]

Interpreting Matrices in Systems

A matrix can represent a system of linear equations.

Example: \[ \begin{bmatrix} 2 & 1\\ 3 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 5\\ 4 \end{bmatrix} \]

Represents: - \(2x + y = 5\)
- \(3x - y = 4\)

Data Tables → Matrices

Matrices often store information like:

Item Price Quantity
A 3 4
B 5 2

Matrix form: \[ \begin{bmatrix} 3 & 4\\ 5 & 2 \end{bmatrix} \]

Strategies

  • Check dimensions before multiplying.
  • Use row-by-column carefully — rewrite rows/columns if needed.
  • Matrix multiplication often encodes compositions of transformations.
  • For systems: rows correspond to equations; columns correspond to variables.
  • When stuck, write out one entry at a time.

Worked Examples

Example 1 — Add matrices

Compute: \[ \begin{bmatrix} 4 & -1\\ 2 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 5\\ 0 & -4 \end{bmatrix} \]

Solution: \[ \begin{bmatrix} 5 & 4\\ 2 & -1 \end{bmatrix} \]

Example 2 — Scalar multiplication

Compute: \[ -3 \begin{bmatrix} 2 & -2\\ 1 & 4 \end{bmatrix} \]

Solution: \[ \begin{bmatrix} -6 & 6\\ -3 & -12 \end{bmatrix} \]

Example 3 — Multiply matrices

Compute: \[ \begin{bmatrix} 2 & 1\\ 1 & 3 \end{bmatrix} \begin{bmatrix} 4 & 0\\ -2 & 5 \end{bmatrix} \]

Work:

First row × first column: \[ 2(4) + 1(-2) = 6 \]

First row × second column: \[ 2(0) + 1(5) = 5 \]

Second row × first column: \[ 1(4) + 3(-2) = -2 \]

Second row × second column: \[ 1(0) + 3(5) = 15 \]

Final result: \[ \begin{bmatrix} 6 & 5\\ -2 & 15 \end{bmatrix} \]

Example 4 — System as a matrix

System: - \(3x + y = 11\)
- \(x - 2y = -3\)

Matrix form: \[ \begin{bmatrix} 3 & 1\\ 1 & -2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 11\\ -3 \end{bmatrix} \]

WarningCommon Mistakes
  • Trying to multiply matrices with incompatible dimensions.
  • Using element-by-element multiplication instead of row-by-column.
  • Forgetting that \(AB \ne BA\).
  • Mixing up rows and columns when forming system matrices.
  • Arithmetic errors in multi-step multiplications.

Practice Problems

  1. Add:
    $$ \[\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\]
  • \[\begin{bmatrix} 4 & 0\\ -1 & 5 \end{bmatrix}\] $$

  1. Scalar multiply:
    \[ 5 \begin{bmatrix} -2 & 3\\ 1 & 4 \end{bmatrix} \]

  2. Multiply:
    \[ \begin{bmatrix} 1 & -1\\ 2 & 3 \end{bmatrix} \begin{bmatrix} 4\\ 5 \end{bmatrix} \]

  3. Write the system represented by:
    \[ \begin{bmatrix} 2 & 3\\ -1 & 4 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 7\\ 10 \end{bmatrix} \]

1.
\[ \begin{bmatrix} 5 & 2\\ 2 & 9 \end{bmatrix} \]

2.
\[ \begin{bmatrix} -10 & 15\\ 5 & 20 \end{bmatrix} \]

3.
Compute row-by-column:
\[ \langle 1,-1 \rangle \cdot \langle 4,5 \rangle = 1(4) + (-1)(5) = -1 \]
\[ \langle 2,3 \rangle \cdot \langle 4,5 \rangle = 2(4) + 3(5) = 23 \]

Answer: \[ \begin{bmatrix} -1\\ 23 \end{bmatrix} \]

4.
System:
\(2x + 3y = 7\)
\(-x + 4y = 10\)

Summary

  • Matrices store data and represent transformations.
  • Add/subtract elementwise; multiply row-by-column.
  • Dimensions must be compatible for multiplication.
  • Systems of equations can be written in matrix form.
  • Dimensions multiply like: \((m \times n)(n \times p)\).
  • If unsure, compute one entry at a time.
  • Matrix multiplication → composition of transformations.
  • Watch for sign errors.