Matrices

Lesson Objectives

Understand different types of matrices and basic operations.
Perform matrix addition, subtraction, multiplication, and scalar multiplication.
Calculate determinants and inverses of matrices.
Solve systems of equations using matrices.

Lesson Introduction

Matrices are rectangular arrays of numbers used to represent and solve systems of equations and transformations in various mathematical and real-world applications. This lesson will guide you through the types of matrices, matrix operations, and solving equations with them.

Worked Example

Matrix Basics and Types

Example 1: Identify the order of the matrix: \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \)
Step 1: Count rows = 2
Step 2: Count columns = 3
Order: 2 × 3

Example 2: Is \( B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) a square matrix?
Yes. Rows = 2, Columns = 2 → Square matrix

Example 3: Identify the type of matrix: \( C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
All diagonal = 1, others = 0 → Identity Matrix

Example 4: Identify the matrix type: \( D = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \)
Off-diagonal elements are negatives → Skew-Symmetric

Example 5: Is \( E = \begin{bmatrix} 5 \end{bmatrix} \) a matrix?
Yes. It is a 1×1 matrix → Scalar Matrix

Matrix Operations

Example 1: Add: \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)

A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

Example 2: Scalar multiplication: 3 × \( C = \begin{bmatrix} 1 & 0 \\ -2 & 4 \end{bmatrix} \)
Multiply each element:

\begin{bmatrix} 3 & 0 \\ -6 & 12 \end{bmatrix}

Example 3: Subtract: \( D = \begin{bmatrix} 9 & 5 \\ 4 & 3 \end{bmatrix}, E = \begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix} \)

D - E = \begin{bmatrix} 5 & 4 \\ 3 & 0 \end{bmatrix}

Example 4: Multiply: \( F = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, G = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \)

FG = \begin{bmatrix} 1×2+2×1 & 1×0+2×2 \\ 3×2+4×1 & 3×0+4×2 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix}

Example 5: Check if \( H = \begin{bmatrix} 2 & -3 \\ 4 & 5 \end{bmatrix} \) is invertible.
Determinant: \( \text{det}(H) = 2×5 - (-3)×4 = 10 + 12 = 22 \neq 0 \)
→ Invertible

Determinant and Inverse of a 2×2 Matrix

Example 1: Find the determinant of \( M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
\( \text{det}(M) = 1×4 - 2×3 = 4 - 6 = -2 \)

Example 2: Inverse of \( N = \begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix} \)
\( \text{det} = 2×4 - 1×7 = 8 - 7 = 1 \)

N^{-1} = \frac{1}{1} \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix} = \begin{bmatrix} 4 & -1 \\ -7 & 2 \end{bmatrix}

Example 3: Check if \( P = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) is invertible.
\( \text{det}(P) = 1×4 - 2×2 = 4 - 4 = 0 \) → Not invertible

Example 4: Inverse of \( Q = \begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix} \)
\( \text{det} = 5×1 - 3×2 = 5 - 6 = -1 \)

Q^{-1} = \frac{1}{-1} \begin{bmatrix} 1 & -3 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix} -1 & 3 \\ 2 & -5 \end{bmatrix}

Example 5: Determinant of \( R = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix} \)
\( \text{det}(R) = 3×1 - 4×2 = 3 - 8 = -5 \)

Exercise

Find the order of the matrix: \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \)
Determine if \( \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \) is a scalar matrix.
Add: \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \)
Subtract: \( \begin{bmatrix} 5 & 4 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \)
Multiply: \( \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} × \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
Find the determinant of: \( \begin{bmatrix} 6 & 4 \\ 3 & 2 \end{bmatrix} \)
Is the matrix \( \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \) invertible?
[NECO] Find the inverse of: \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) (Past Question)
Determine the type of matrix: \( \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \)
Calculate the determinant of: \( \begin{bmatrix} 7 & 2 \\ 5 & 1 \end{bmatrix} \)

Conclusion/Recap

Matrices are essential for organizing and solving mathematical problems involving systems of equations, transformations, and more. Understanding matrix types, operations, and inverses helps develop a strong foundation in higher-level math applications.