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Matrix Operations and Applications

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Matrix Operations and Applications - Complete Interactive Lesson

Part 1: Matrix Basics: Shape, Entries, and Equality

🔢 Matrix Operations and Applications

Part 1 of 7 — Matrix Basics: Shape, Entries, and Equality

Topics in This Part

Section
What Is a Matrix?
Dimensions and Entry Notation
When Are Two Matrices Equal?

🔑 Key Concept: A matrix is a rectangular grid of numbers. Before we can add, multiply, or invert matrices, we need a precise language for their shape (rows × columns) and their entries. That vocabulary is Part 1.

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns, written inside brackets:

$A = \begin{bmatrix} 2 & -1 & 0 \\ 4 & 5 & 3 \end{bmatrix}$

Rows run horizontally (this $A$ has $2$ rows).
Columns run vertically (this $A$ has $3$ columns).

The dimensions (or size) of a matrix are written $\text{rows} \times \text{columns}$ . Here $A$ is a $2 \times 3$ matrix.

🔑 Order matters: Always say rows first, columns second. A $2 \times 3$ matrix is not the same shape as a $3 \times 2$ matrix.

Special shapes

Name	Shape	Example
Row matrix	$1 \times n$	$\begin{bmatrix} 3 & 7 \end{bmatrix}$

Concept Check 🎯

Entry Notation: $a_{ij}$

We name a matrix with a capital letter and refer to a single number inside it as an entry (or element). The entry in row $i$ , column $j$ of matrix $A$ is written

Locate the Entries 🧮

Use the matrix $B = \begin{bmatrix} 7 & -2 & 9 \\ 0 & 6 & -4 \end{bmatrix}$

When Are Two Matrices Equal?

Two matrices are equal when both of these hold:

They have the same dimensions, and
Every corresponding entry is equal: $a_{ij} = b_{ij}$ for all $i, j$ .

Concept Check 🎯

Part 2: Addition, Subtraction, and Scalar Multiplication

🔢 Matrix Operations and Applications

Part 2 of 7 — Addition, Subtraction, and Scalar Multiplication

🔑 The Idea: Adding, subtracting, and scaling matrices all happen entry-by-entry. These are the easy operations — the only rule to watch is that addition and subtraction require matching dimensions.

Adding and Subtracting Matrices

To add (or subtract) two matrices of the same size, add (or subtract) corresponding entries:

$[\begin{array}{cc} a & b \\ c & d \end{array}] + [\begin{array}{cc} e & f \\ g & h \end{array}] = [\begin{array}{cc} a + e & b + f \end{array}$

Part 3: Matrix Multiplication

🔢 Matrix Operations and Applications

Part 3 of 7 — Matrix Multiplication

🔑 The Big Idea: Matrix multiplication is not entry-by-entry. Each entry of the product is a row dotted with a column. This row-by-column rule is the engine behind transformations, systems, and applications later in the lesson.

The Row-by-Column Rule

To find the entry in row $i$ , column $j$ of the product $AB$ , take row $i$ of , , multiply matching numbers, and add.

Part 4: The Identity Matrix and Inverses

🔢 Matrix Operations and Applications

Part 4 of 7 — The Identity Matrix and Inverses

🔑 The Idea: The identity matrix $I$ plays the role of " $1$ " for matrices, and an inverse $A^{-1}$ plays the role of " $\frac{1}{A}$ ." Inverses are the key to — and to solving systems in Part 6.

Part 5: Determinants

🔢 Matrix Operations and Applications

Part 5 of 7 — Determinants

🔑 The Idea: The determinant is a single number that captures a square matrix's "size." It tells you instantly whether a matrix is invertible, and it powers a shortcut for solving systems (Cramer's Rule).

Determinant of a $2 \times 2$ Matrix

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is

Part 6: Solving Systems with Matrices

🔢 Matrix Operations and Applications

Part 6 of 7 — Solving Systems with Matrices

🔑 The Payoff: A linear system can be packed into a single equation $AX = B$ . Multiplying by $A^{-1}$ solves it in one shot: $X = A^{-1}B$ .

Part 7: Applications, Mixed Practice & Exit Quiz

🔢 Matrix Operations and Applications

Part 7 of 7 — Applications, Mixed Practice & Exit Quiz

You can now describe matrices, add and scale them, multiply them, invert them, take determinants, and solve systems. This final part shows two real applications — transformations and networks — then a mastery check.

Application 1: Transformations of Points

Multiplying a point $\begin{bmatrix} x \\ y \end{bmatrix}$ by a $2\times2$ matrix moves it in the plane. A few famous transformation matrices:

The first subscript is the row; the second is the column.

$A = \begin{bmatrix} 2 & -1 & 0 \\ 4 & 5 & 3 \end{bmatrix}$

$a_{11} = 2$ (row $1$ , column $1$ )
$a_{13} = 0$ (row $1$ , column $3$ )
$a_{23} = 3$ (row $2$ , column $3$ )
$a_{21} = 4$ (row $2$ , column $1$ )

💡 Read the subscripts like coordinates, but (row, column) instead of $(x, y)$ .

1) $b_{12} = \,?$ 2) $b_{21} = \,?$ 3) $b_{23} = \,?$

$\begin{bmatrix} x & 3 \\ -1 & y \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ -1 & 8 \end{bmatrix}$

Matching entry-by-entry forces $x = 5$ and $y = 8$ .

⚠️ A $2 \times 3$ matrix can never equal a $3 \times 2$ matrix, even if they contain the same numbers — the shapes differ. Equality of matrices is a powerful tool: it turns one matrix equation into a system of equations, one per entry.

[a c b d] + [e g f h] = [a + e c + g b + f d + h]

$\begin{bmatrix} 2 & -1 \\ 0 & 5 \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 6 & -2 \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 6 & 3 \end{bmatrix}$

$\begin{bmatrix} 7 & 8 \\ 1 & 0 \end{bmatrix} - \begin{bmatrix} 2 & 5 \\ 4 & -3 \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ -3 & 3 \end{bmatrix}$

⚠️ Dimensions must match. You cannot add a $2 \times 2$ matrix to a $2 \times 3$ matrix — the sum is undefined.

Scalar Multiplication

A scalar is just a single number. To multiply a matrix by a scalar $k$ , multiply every entry by $k$ :

$k\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix}$

Worked Example

$3\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$

This lets us combine operations. For example, $2A - B$ means "double every entry of $A$ , then subtract $B$ entry-by-entry."

$2 [\begin{array}{cc} 1 & 3 \\ 5 & 0 \end{array}] - [\begin{array}{cc} 4 & 1 \\ 2 & - 3 \end{array}] = [\begin{array}{cc} 2 & 6 \\ 10 & 0 \end{array}] - [\begin{array}{cc} 4 & 1 \\ 2 & - 3 \end{array}] = [\begin{array}{cc} - 2 & 5 \end{array}$

Concept Check 🎯

Compute $2A - B$ 🧮

Let $A = \begin{bmatrix} 3 & -1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ -2 & 5 \end{bmatrix}$ .

Compute $2A - B$ and enter its entries reading left to right, top to bottom.

Entry $(1,1) = \,?$ Entry $(1,2) = \,?$ Entry $(2,1) = \,?$ Entry

Defined or Not? 🔽

For each expression, decide whether the operation is defined (legal) given the stated shapes.

column $j$ of $B$

For $2 \times 2$ matrices:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{bmatrix}$

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Entry $(1,1)$ : row $1$ $\begin{bmatrix}1 & 2\end{bmatrix}$ · column $1$ $\begin{bmatrix}5 \\ 7\end{bmatrix} = 1\cdot5 + 2\cdot7 = 19$
Entry $(1,2)$ : $1\cdot6 + 2\cdot8 = 22$
Entry $(2,1)$ : $3\cdot5 + 4\cdot7 = 43$
Entry $(2,2)$ : $3\cdot6 + 4\cdot8 = 50$

$= \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

When Is $AB$ Defined? What Size Is It?

The product $AB$ exists only when the inner dimensions match:

$\underbrace{(m \times n)}_{A}\ \underbrace{(n \times p)}_{B} \;=\; (m \times p)$

The inner numbers ( $n$ and $n$ ) must be equal — that is what lets a row of $A$ pair with a column of $B$ .
The outer numbers ( $m$ and $p$ ) give the .

Example

A $2 \times 3$ matrix times a $3 \times 4$ matrix: inner $3 = 3$ ✓, so the product exists and is $2 \times 4$ .

⚠️ Order matters! In general $AB \ne BA$ . Matrix multiplication is not commutative. Sometimes $AB$ is defined while $BA$ is not even a legal product.

Is It Defined? What Size? 🔽

For each product, decide whether it is defined, and if so, give its dimensions.

Concept Check 🎯

Multiply Them 🧮

Compute the product $\begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ and enter its entries left to right, top to bottom.

Entry $(1,1) = \,?$ Entry $(1,2) = \,?$ Entry $(2,1) = \,?$ Entry

The Identity Matrix

The identity matrix $I$ is a square matrix with $1$ 's on the main diagonal and $0$ 's everywhere else:

$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

It behaves like the number $1$ : for any compatible matrix $A$ ,

$AI = IA = A$

Quick check

$\begin{bmatrix} 7 & 2 \\ -3 & 5 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ -3 & 5 \end{bmatrix}$

Multiplying by $I$ leaves a matrix unchanged.

What Is an Inverse?

The inverse of a square matrix $A$ , written $A^{-1}$ , is the matrix that "undoes" $A$ :

$A A^{-1} = A^{-1} A = I$

For a $2 \times 2$ matrix there is a clean formula. If

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \qquad A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

Three moves to remember:

Swap $a$ and $d$ (the diagonal entries).
Negate $b$ and $c$ (the off-diagonal entries).
Divide the whole thing by $ad - bc$ .

⚠️ The number $ad - bc$ is the determinant (Part 5). If $ad - bc = 0$ , you would be dividing by zero — the matrix has no inverse and is called singular.

Worked Example

Find the inverse of $A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}$ .

Step 1 — determinant: $ad - bc = (4)(4) - (3)(5) = 16 - 15 = 1$ .

Step 2 — swap, negate, divide:

$A^{-1} = \frac{1}{1}\begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix}$

Step 3 — verify:

$A A^{- 1} = [\begin{array}{cc} 4 & 3 \\ 5 & 4 \end{array}] [\begin{array}{cc} 4 & - 3 \\ - 5 & 4 \end{array}] = [\begin{array}{cc} 16 - 15 & - 12 + 12 \\ 20 - 20 & - 15 + 16 \end{array}] = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = I$

Build the Inverse 🔽

You are inverting $A = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$ . Choose each piece.

Concept Check 🎯

Find the Inverse 🧮

Find $A^{-1}$ for $A = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}$ , then enter its entries left to right, top to bottom.

Entry $(1,1) = \,?$ Entry $(1,2) = \,?$ Entry $(2,1) = \,?$ Entry

$\det(A) = \lvert A \rvert = ad - bc$

It is the product of the main diagonal minus the product of the anti-diagonal.

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

$\det\begin{bmatrix} 5 & 6 \\ -2 & 1 \end{bmatrix} = (5)(1) - (6)(-2) = 5 + 12 = 17$

🔑 Invertibility test: $A$ has an inverse if and only if $\det(A) \ne 0$ .

Compute Determinants 🧮

Find each $2\times2$ determinant.

1) $\det\begin{bmatrix} 6 & 2 \\ 5 & 3 \end{bmatrix} = \,?$ 2) $\det\begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix} = \,?$ 3) $\det\begin{bmatrix} 3 & 6 \\ 1 & 2 \end{bmatrix} = \,?$

Determinant of a $3 \times 3$ Matrix (Expansion)

For a $3 \times 3$ matrix we expand along the top row. Each top entry multiplies the determinant of the $2 \times 2$ matrix left after deleting that entry's row and column, with a + − + sign pattern:

$\det\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a\,(ei - fh) - b\,(di - fg) + c\,(dh - eg)$

Worked Example

$\det\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}$

$= 1(4\cdot6 - 5\cdot0) - 2(0\cdot6 - 5\cdot1) + 3(0\cdot0 - 4\cdot1)$

$= 1(24) - 2(-5) + 3(-4) = 24 + 10 - 12 = 22$

💡 Watch the middle sign: the $b$ -term is subtracted. The pattern is $+\ -\ +$ .

Concept Check 🎯

Expand a $3\times3$ 🧮

Compute, expanding along the top row: $\det\begin{bmatrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ 2 & 0 & 1 \end{bmatrix} = \,?$

Writing a System as $AX = B$

The system $\begin{cases} 2x + 3y = 8 \\ x + 2y = 5 \end{cases}$ becomes the matrix equation

$\underbrace{\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}}_{A}\underbrace{\begin{bmatrix} x \\ y \end{bmatrix}}_{X} = \underbrace{\begin{bmatrix} 8 \\ 5 \end{bmatrix}}_{B}$

$A$ is the coefficient matrix (the numbers in front of $x$ and $y$ ).
$X$ is the variable matrix.
$B$ is the constant matrix (right-hand sides).

💡 Multiplying $A$ by $X$ reproduces the left sides of both equations — that is why this encoding works.

Solving with the Inverse: $X = A^{-1}B$

Starting from $AX = B$ , multiply both sides on the left by $A^{-1}$ :

$A^{-1}AX = A^{-1}B \;\Rightarrow\; IX = A^{-1}B \;\Rightarrow\; X = A^{-1}B$

Worked Example

For $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ , , so

$A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$

Then

$X = A^{-1}B = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}\begin{bmatrix} 8 \\ 5 \end{bmatrix} = \begin{bmatrix} 16 - 15 \\ -8 + 10 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

So $x = 1$ , $y = 2$ .

✅ Check: $2(1) + 3(2) = 8$ ✓ and $1 + 2(2) = 5$ ✓.

Walk Through the Solve 🔽

You are solving $\begin{cases} x + y = 5 \\ 2x + 3y = 13 \end{cases}$ with $A = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$ .

Solve the System 🧮

Use $X = A^{-1}B$ to solve $\begin{cases} 3x + 2y = 7 \\ x + y = 3 \end{cases}$

Here $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$ .

1) $x = \,?$ 2) $y = \,?$

Concept Check 🎯

Transformation	Matrix
Scale by $k$	$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$
Reflect over $x$ -axis	$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
Rotate $90°$ counterclockwise	$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

Example: rotate $(3, 1)$ by $90°$ CCW

$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0\cdot3 + (-1)\cdot1 \\ 1\cdot3 + 0\cdot1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$

So $(3, 1) \to (-1, 3)$ .

Application 2: Inventory and Networks

Matrices organize real data and let you combine it with multiplication.

Cost example. A store stocks two products with a price vector and a quantity vector. Suppose Store A sold $\begin{bmatrix} 4 & 6 \end{bmatrix}$ units (product 1, product 2) at prices $\begin{bmatrix} 5 \\ 3 \end{bmatrix}$ dollars:

$\begin{bmatrix} 4 & 6 \end{bmatrix}\begin{bmatrix} 5 \\ 3 \end{bmatrix} = 4\cdot5 + 6\cdot3 = 20 + 18 = 38$

Total revenue $= 38$ , i.e. $38. A single $(1\times2)(2\times1)$ product summed everything at once.

💡 The same idea scales up: an $m \times n$ "sales" matrix times an $n \times 1$ "price" vector gives total revenue per store in one multiplication.

Application Check 🎯

Mixed Mastery 🧮

1) $\det\begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix} = \,?$ 2) Rotating $(0, 2)$ by $90°$ CCW with $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ gives the point $(\,?,\ \,?)$ . Enter the $x$ -coordinate, then the $y$ -coordinate.

Exit Quiz ✅

Answer all three to finish the lesson.

2 [1530] - [42 1 - 3] = [21060] - [42 1 - 3] = [- 2 8 53]

size of the product

ad−bc1[d−c−ba]

A A^{- 1} = [4534] [4 - 5 - 3 4] = [16 - 15 20 - 20 - 12 + 12 - 15 + 16] = [1001] = I ✓

\det(A) = (2)(2)-(3)(1) = 1

det (A) = (2) (2) - (3) (1) = 1

Matrix Operations and Applications

Matrix Operations and Applications - Complete Interactive Lesson

Part 1: Matrix Basics: Shape, Entries, and Equality

🔢 Matrix Operations and Applications

Topics in This Part

What Is a Matrix?

Special shapes

Entry Notation: aija_{ij}aij​

When Are Two Matrices Equal?

Part 2: Addition, Subtraction, and Scalar Multiplication

🔢 Matrix Operations and Applications

Adding and Subtracting Matrices

Part 3: Matrix Multiplication

🔢 Matrix Operations and Applications

The Row-by-Column Rule

Part 4: The Identity Matrix and Inverses

🔢 Matrix Operations and Applications

Part 5: Determinants

🔢 Matrix Operations and Applications

Determinant of a 2×22 \times 22×2 Matrix

Part 6: Solving Systems with Matrices

🔢 Matrix Operations and Applications

Part 7: Applications, Mixed Practice & Exit Quiz

🔢 Matrix Operations and Applications

Application 1: Transformations of Points

Example

Example

Worked Example

Scalar Multiplication

Worked Example

Worked Example

When Is ABABAB Defined? What Size Is It?

Example

The Identity Matrix

Quick check

What Is an Inverse?

Worked Example

Examples

Determinant of a 3×33 \times 33×3 Matrix (Expansion)

Worked Example

Writing a System as AX=BAX = BAX=B

Solving with the Inverse: X=A−1BX = A^{-1}BX=A−1B

Worked Example

Example: rotate (3,1)(3, 1)(3,1) by 90°90°90° CCW

Application 2: Inventory and Networks

Entry Notation: $a_{ij}$

Determinant of a $2 \times 2$ Matrix

When Is $AB$ Defined? What Size Is It?

Determinant of a $3 \times 3$ Matrix (Expansion)

Writing a System as $AX = B$

Solving with the Inverse: $X = A^{-1}B$

Example: rotate $(3, 1)$ by $90°$ CCW