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AP Precalculus: Matrices

Master matrix operations, determinants, inverses, and transformations

🔢 Operations ✖️ Multiplication 🔄 Inverses 📐 Transformations

📚 Understanding Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are essential tools for solving systems of equations, describing transformations, and organizing data. This guide covers vocabulary, operations, determinants, inverses, and geometric transformations.

1 Matrix Vocabulary

A matrix is a rectangular array of numbers. The dimensions are written as \(m \times n\) (rows × columns).

Element \(a_{ij}\)
Entry in row \(i\), column \(j\)
Square Matrix
\(n \times n\) — same rows and columns
Row Matrix
\(1 \times n\) — single row
Column Matrix
\(m \times 1\) — single column
Zero Matrix
All entries are 0
Main Diagonal
Entries \(a_{11}, a_{22}, a_{33}, ...\)

Identity Matrix \(I_n\)

2×2 Identity
\(I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
3×3 Identity
\(I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
💡 Identity Property

For any matrix \(A\): \(AI = IA = A\). The identity matrix acts like multiplying by 1.

2 Matrix Addition & Subtraction

To add or subtract matrices, they must have the same dimensions. Add or subtract corresponding elements.

Element-wise Operations \((A + B)_{ij} = a_{ij} + b_{ij}\)     \((A - B)_{ij} = a_{ij} - b_{ij}\)
📌 Example

\(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 1 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 2+5 & 3+1 \\ 1+2 & 4+3 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 3 & 7 \end{pmatrix}\)

⚠️ Dimension Requirement

You cannot add a 2×3 matrix to a 3×2 matrix. Dimensions must match exactly!

3 Scalar Multiplication

Scalar multiplication multiplies every element of a matrix by a constant (scalar).

Scalar Multiplication Formula \((cA)_{ij} = c \cdot a_{ij}\)
📌 Example

\(3 \cdot \begin{pmatrix} 2 & -1 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 6 & -3 \\ 12 & 0 \end{pmatrix}\)

Combined Operations

📌 Example: Linear Combination

Find \(2A - 3B\) where \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 1 \\ 2 & 1 \end{pmatrix}\)

\(2A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}\), \(3B = \begin{pmatrix} 0 & 3 \\ 6 & 3 \end{pmatrix}\)

\(2A - 3B = \begin{pmatrix} 2-0 & 4-3 \\ 6-6 & 8-3 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 5 \end{pmatrix}\)

4 Matrix Multiplication

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. The result has dimensions from the outer numbers.

Dimension Rule \(A_{m \times \boxed{n}} \cdot B_{\boxed{n} \times p} = C_{m \times p}\)
Element Formula \((AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}\)
📌 Example: 2×2 Multiplication

\(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}\)

Row 1 × Col 1: \(1(5) + 2(7) = 5 + 14 = 19\)

Row 1 × Col 2: \(1(6) + 2(8) = 6 + 16 = 22\)

Row 2 × Col 1: \(3(5) + 4(7) = 15 + 28 = 43\)

Row 2 × Col 2: \(3(6) + 4(8) = 18 + 32 = 50\)

Result: \(\begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}\)

⚠️ Not Commutative!

Matrix multiplication is NOT commutative: \(AB \neq BA\) in general. Order matters!

5 Properties of Matrix Operations

Matrix operations follow specific algebraic properties similar to (but not identical to) number properties.

📦 Commutative (Addition)
\(A + B = B + A\)
Order doesn't matter for addition
📦 Associative
\(A + (B + C) = (A + B) + C\)
\(A(BC) = (AB)C\)
Grouping doesn't matter
📦 Distributive
\(A(B + C) = AB + AC\)
\((A + B)C = AC + BC\)
Distribute multiplication over addition
📦 Identity
\(AI = IA = A\)
Identity matrix acts like 1
Transpose Rule \((AB)^T = B^T A^T\) — Note the order reversal!

6 Determinant of a Matrix

The determinant is a scalar value calculated from a square matrix. It determines whether a matrix is invertible and has geometric meaning (area/volume scaling).

2×2 Determinant

\(\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\)

"Main diagonal minus off diagonal"

3×3 Determinant

Expand by minors (cofactors) along any row or column, OR use Sarrus' Rule for diagonals

📌 Example: 2×2

\(\det\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} = (3)(4) - (2)(1) = 12 - 2 = 10\)

📌 Example: 3×3 (Expansion along Row 1)

\(\det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\)

\(= 1 \cdot \det\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} - 2 \cdot \det\begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} + 3 \cdot \det\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix}\)

\(= 1(45-48) - 2(36-42) + 3(32-35)\)

\(= 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0\)

💡 Invertibility Check

A matrix is invertible if and only if its determinant is NOT zero: \(\det(A) \neq 0\).

7 Inverse of a Matrix

The inverse of matrix \(A\), denoted \(A^{-1}\), satisfies \(AA^{-1} = A^{-1}A = I\). Only square matrices with non-zero determinant have inverses.

2×2 Inverse Formula \[A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\]

Steps for 2×2 Inverse

  1. Calculate the determinant \(ad - bc\)
  2. Check that determinant \(\neq 0\) (otherwise, no inverse exists)
  3. Swap the main diagonal elements (\(a\) and \(d\))
  4. Change signs of off-diagonal elements (\(b\) and \(c\))
  5. Multiply by \(\frac{1}{\det}\)
📌 Example

Find the inverse of \(A = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}\)

Determinant: \((4)(2) - (3)(3) = 8 - 9 = -1\)

Inverse: \(A^{-1} = \frac{1}{-1} \begin{pmatrix} 2 & -3 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} -2 & 3 \\ 3 & -4 \end{pmatrix}\)

3×3 Inverse Formula \(A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\)
where adj(A) is the adjugate (transpose of cofactor matrix)

8 Solving Matrix Equations

Matrix equations can be solved using the inverse matrix, similar to dividing both sides by the coefficient in algebra.

Solving \(AX = B\) If \(A\) is invertible: \(X = A^{-1}B\)
📌 Example: Solve System Using Inverse

System: \(\begin{cases} 2x + 3y = 7 \\ x + 2y = 4 \end{cases}\)

Matrix form: \(\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}\)

Find inverse: \(\det = 4 - 3 = 1\)

\(A^{-1} = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix}\)

Solve: \(\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 7 \\ 4 \end{pmatrix} = \begin{pmatrix} 14-12 \\ -7+8 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\)

Solution: \(x = 2\), \(y = 1\)

9 Transformation Matrices

Transformation matrices perform geometric transformations on points. To transform point \((x, y)\), multiply: \(\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ y \end{pmatrix}\)

Reflection over x-axis
\(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
Reflection over y-axis
\(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
Reflection over \(y = x\)
\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
Rotation by \(\theta\) (CCW)
\(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)
Scale by factor \(k\)
\(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
90° Rotation (CCW)
\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
📌 Example: Rotate 90° CCW

Rotate point \((3, 2)\) by 90° counterclockwise:

\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 0(3) + (-1)(2) \\ 1(3) + 0(2) \end{pmatrix} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

💡 Vertex Matrix

To transform a shape, put all vertices as columns in a matrix and multiply by the transformation matrix. Each column of the result is a transformed vertex!

📋 Quick Reference

Addition

Same dimensions required

Multiplication

Inner dimensions must match

2×2 Determinant

\(ad - bc\)

2×2 Inverse

\(\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

Invertible

\(\det(A) \neq 0\)

Solve \(AX = B\)

\(X = A^{-1}B\)

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