FAD1015 L29-L30 — Matrices (Inverse & Systems of Equations)

Lectures 29–30 covering matrix inverses, determinants, and solving systems of linear equations using matrix methods. Source file: (L29L30) - WEEK 17_MATRICES.pdf (35 slides)

Summary

This lecture covers inverse matrices (definition, 2×2 formula, adjoint method, elementary row operations) and three methods for solving systems of linear equations with three variables: inverse matrix method, Gauss-Jordan elimination, and Cramer's Rule.

L29: Inverse Matrices

1. Definition of Inverse Matrices

Let $A$ be any square matrix of order $n \times n$. If there exists a matrix $B$ such that:

$$AB = BA = I$$

Then $B$ is called the inverse of $A$ and vice-versa. The inverse of $A$ is denoted by $A^{-1}$.

Also:

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

Example 1: If $A = \begin{pmatrix} 2 & 5 \ 1 & 3 \end{pmatrix}$, $B = \begin{pmatrix} 3 & -5 \ -1 & 2 \end{pmatrix}$, show that $A$ is the inverse matrix of $B$.

(Solution: verify $AB = BA = I$)

2. Inverse of a 2 × 2 Matrix

Let $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ and $B = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$

Then $BA = (ad - bc)\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$

Hence, if $ad - bc \neq 0$:

$$A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \quad \text{where } |A| = ad - bc$$

Singular vs Non-Singular:

  • If $A^{-1}$ exists, $A$ is called a non-singular matrix
  • When $|A| = 0$, $A^{-1}$ does not exist and $A$ is called a singular matrix

Steps to find inverse of 2×2:

  1. Interchange the elements of the leading diagonal
  2. Reverse the sign of the other elements
  3. Divide the matrix by the determinant

Example 2: Find the inverse of $A = \begin{pmatrix} 3 & 1 \ 4 & 2 \end{pmatrix}$

Example 3: If $A = \begin{pmatrix} 1 & 0 \ 4 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & -1 \ 1 & 0 \end{pmatrix}$, find $AB$ and show that $(AB)^{-1} = B^{-1}A^{-1}$

Example 4: Given $A = \begin{pmatrix} -2 & 1 & 1 \ 3 & -2 & 1 \ -1 & 2 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 3 & 3 \ 2 & 3 & 5 \ 4 & 3 & 1 \end{pmatrix}$. Find $AB$. Hence deduce the inverse of $A$.

3. Adjoint Matrix

If $A$ is a $n \times n$ matrix and $C_{ij}$ is the cofactor for element $a_{ij}$, then the cofactor matrix of $A$ is:

$$C = \begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \ C_{21} & C_{22} & \cdots & C_{2n} \ \vdots & \vdots & \ddots & \vdots \ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$$

The adjoint of matrix $A$ is:

$$\text{adj}(A) = C^T$$

Example 5: Find adjoint of $A$ if $A = \begin{pmatrix} 2 & -1 & 3 \ 0 & 3 & 1 \ 1 & 0 & 2 \end{pmatrix}$

4. Finding Inverse Using Matrix Adjoint

Given a non-singular square matrix $A$ of any size:

$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$

Example 6: Find the inverse of $A = \begin{pmatrix} 2 & 3 & 0 \ -5 & 0 & 4 \ 0 & 2 & 1 \end{pmatrix}$

5. Elementary Row Operations (ERO)

There are three elementary row operations:

  1. Interchange any two rows ($R_i \leftrightarrow R_j$)
  2. Multiply all elements of a row by a scalar ($R_i \rightarrow kR_i$)
  3. Multiply all elements of a row by a scalar and add to another row ($R_j \rightarrow kR_i + R_j$)

When a matrix $A$ is changed into another matrix $B$ after using ERO, $A$ and $B$ are said to be equivalent.

6. Finding Inverse Using ERO

To find the inverse of matrix $A$, start by writing the matrix in the form $(A|I)$ and change it by ERO into $(I|B)$. The resulting matrix $B$ is $A^{-1}$.

$$(A|I) \text{ reduce to } (I|A^{-1})$$

5-step process for 3×3:

  • Step 1: Obtain a 1 in the first position on the leading diagonal
  • Step 2: Obtain zeros under the 1 in the first column
  • Step 3: Obtain a 1 in the second position on the leading diagonal
  • Step 4: Obtain zeros under the 1 in the second column
  • Step 5: Obtain a 1 in the third position on the leading diagonal. Finally obtain zeros above all the 1's

Example 7: Given $A = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{pmatrix}$. Find $A^{-1}$ using ERO.

Example 8: If $A = \begin{pmatrix} 2 & -1 & 1 \ 1 & -1 & -1 \ 2 & -2 & -1 \end{pmatrix}$, find $A^{-1}$ using ERO.

L30: Systems of Linear Equations with Three Variables

1. System of Linear Equations in Matrix Notation

The system: $$\begin{aligned} a_{11}x + a_{12}y + a_{13}z &= b_1 \ a_{21}x + a_{22}y + a_{23}z &= b_2 \ a_{31}x + a_{32}y + a_{33}z &= b_3 \end{aligned}$$

Can be represented as $AX = B$:

$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} b_1 \ b_2 \ b_3 \end{pmatrix}$$

Where $A$ is the coefficients matrix, $X$ is the matrix of variables, and $B$ is the constants matrix.

Example 9: The equations $2x - y + 3z = 4$, $x + 2y - z = 1$ and $3x + y - 2z = 2$ can be written as $AX = B$.

2. Method Using the Inverse Matrix

$$AX = B$$ $$A^{-1}(AX) = A^{-1}B$$ $$(A^{-1}A)X = A^{-1}B$$ $$IX = A^{-1}B$$ $$X = A^{-1}B$$

Note: This method cannot be used if $A$ is a singular matrix ($|A| = 0$).

Example 10: Solve the system: $$\begin{aligned} x + 2y + 7z &= 1 \ x + 3y &= 2 \ -y + 8z &= 3 \end{aligned}$$

3. Gauss-Jordan Elimination Method

Steps for solving $AX = B$:

  1. Write the system in the form $AX = B$
  2. Form an augmented matrix $(A|B)$
  3. Use ERO to reduce $(A|B)$ to $(I|X)$

Possible Solutions:

  1. If $|A| \neq 0$: unique solution given by $X = A^{-1}B$
  2. If $|A| = 0$ and $(\text{adj } A)B = 0$: infinitely many solutions (e.g., row reduces to $(0 \quad 0 \quad 0 ;|; 0)$)
  3. If $|A| = 0$ and $(\text{adj } A)B \neq 0$: no solution (e.g., row reduces to $(0 \quad 0 \quad 0 ;|; c)$ where $c$ is a constant)

Example 11: Solve by Gauss-Jordan elimination: $$\begin{aligned} x + z &= 5 \ 3x + y + 2z &= 11 \ 3x + 2y + 2z &= 10 \end{aligned}$$

4. Cramer's Rule

Cramer's Rule uses determinants to solve systems with the same number of equations as variables.

For 2×2: $$x = \frac{\begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix}}, \quad y = \frac{\begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix}}$$

For 3×3: $$x = \frac{\begin{vmatrix} d_1 & b_1 & c_1 \ d_2 & b_2 & c_2 \ d_3 & b_3 & c_3 \end{vmatrix}}{|A|}, \quad y = \frac{\begin{vmatrix} a_1 & d_1 & c_1 \ a_2 & d_2 & c_2 \ a_3 & d_3 & c_3 \end{vmatrix}}{|A|}, \quad z = \frac{\begin{vmatrix} a_1 & b_1 & d_1 \ a_2 & b_2 & d_2 \ a_3 & b_3 & d_3 \end{vmatrix}}{|A|}$$

Example 12: Solve using Cramer's Rule: $$\begin{aligned} x + y - z &= 0 \ 2x - 3y + z &= 1 \ 2x + y + 2z &= 7 \end{aligned}$$

Example 13 (Word Problem): Aishah bought a marker pen, two examination pads and a file for RM36. Balqis bought three marker pens, two examination pads and a file for RM43 while Camelia bought three marker pens, an examination pad and four files for RM74. Let $x, y, z$ represent prices. Write the system, express as matrix, and solve using Cramer's Rule.

Key Concepts

  • Matrices — Matrix algebra, types, and operations
  • Matrix Inverse — $A^{-1}$ such that $AA^{-1} = I$
  • Determinant — Scalar value $|A|$ for square matrices
  • Adjoint Matrix — Transpose of cofactor matrix
  • Elementary Row Operations — ERO for matrix reduction
  • Gauss-Jordan Elimination — Solving systems via augmented matrix
  • Cramer's Rule — Determinant-based solution method
  • Systems of Linear Equations — $AX = B$ representation

Related Topics

Related Course Page

  • FAD1015 - Mathematics III