Matrices and Determinants

Grade 11 Math Lesson - Matrices and Determinants

Lesson Objectives

Identify different types of matrices and their properties.
Calculate determinants of 2x2 and 3x3 matrices.
Solve systems of linear equations using matrices.
Understand real-world applications of matrices and determinants.

Lesson Introduction

Matrices are powerful tools used in organizing and solving data problems, while determinants provide important properties about matrices. Applications span from solving simultaneous equations to engineering and computer graphics.

Core Lesson Content

Types of Matrices:

Row Matrix: A matrix with only one row.
Column Matrix: A matrix with only one column.
Square Matrix: A matrix with the same number of rows and columns.
Diagonal Matrix: A square matrix where non-diagonal elements are zero.
Identity Matrix: A diagonal matrix with ones on the main diagonal.
Zero Matrix: A matrix with all elements zero.

Determinants:

For a 2x2 matrix:

\( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

Determinant, \( \text{det}(A) = ad - bc \)

For a 3x3 matrix:

\( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \)

Determinant:

\( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \)

Solving Systems of Equations using Matrices:

Using matrices, simultaneous equations can be solved through methods like finding the inverse matrix:

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

Worked Example

Example 1: Identify the type of matrix \( \begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix} \)

Since it has 2 rows and 2 columns, it is a square matrix.

Example 2: Find the determinant of \( \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \)

Use the formula:

\( \text{det} = (4 \times 6) - (7 \times 2) \) \( \text{det} = 24 - 14 = 10 \)

Example 3: Find the determinant of \( \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \)

Expanding along the first row:

\( \text{det} = 1(4 \times 6 - 5 \times 0) - 2(0 \times 6 - 5 \times 1) + 3(0 \times 0 - 4 \times 1) \) \( = 1(24 - 0) - 2(0 - 5) + 3(0 - 4) \) \( = 24 + 10 - 12 \) \( = 22 \)

Example 4: Solve \( 2x + y = 5 \), \( x - y = 1 \) using matrices.

Matrix form:

\( A = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 1 \end{bmatrix} \)

Find \( A^{-1} \) and compute \( X = A^{-1}B \).

First, \( \text{det}(A) = (2 \times -1) - (1 \times 1) = -2 -1 = -3 \)

Inverse of A:

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

Multiplying:

\( X = A^{-1}B = \frac{1}{-3} \begin{bmatrix} (-1)(5) + (-1)(1) \\ (-1)(5) + (2)(1) \end{bmatrix} \) \( X = \frac{1}{-3} \begin{bmatrix} -6 \\ -3 \end{bmatrix} \) \( = \begin{bmatrix} 2 \\ 1 \end{bmatrix} \)

Thus, \( x = 2 \), \( y = 1 \).

Example 5: What is the determinant of the identity matrix \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

Using the formula:

\( \text{det} = (1 \times 1) - (0 \times 0) = 1 - 0 = 1 \)

Exercises

Identify the type of matrix: \( \begin{bmatrix} 2 & 4 & 6 \end{bmatrix} \)
Find the determinant of \( \begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix} \)
[WAEC] Find the determinant of \( \begin{bmatrix} 7 & 2 \\ 4 & 3 \end{bmatrix} \). [Past Question]
Find the determinant of \( \begin{bmatrix} 2 & 0 & 1 \\ 3 & 0 & 0 \\ 5 & 1 & 1 \end{bmatrix} \)
[NABTEC] Solve the system \( 3x + 4y = 10 \), \( x - 2y = -1 \) using matrices. [Past Question]
[NECO] Solve \( x + y = 6 \), \( x - y = 2 \) using matrices. [Past Question]
Find the determinant of \( \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} \)
Determine whether \( \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) has an inverse.
[JAMB] Solve for \( x \) and \( y \): \( 2x + 3y = 7 \), \( x - y = 2 \). [Past Question]
Find the inverse of \( \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} \) if it exists.

Conclusion/Recap

In this lesson, we explored different types of matrices, how to find their determinants, and how to solve systems of equations using matrices. Mastery of these skills is crucial for deeper understanding in areas such as linear transformations and advanced algebra. Up next: Linear Inequalities.