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6. Matrices
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VCE 11 General 2023

6.08 Inverse matrices

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Introduction

In mathematics, when you multiply a number by its inverse the result is 1. In other words, a number's inverse is the unique number which results in1 when the two are multiplied together. In matrices, the product of a matrix and its inverse (if it exists) is the identity matrix, I.

With real numbers, a number's inverse is its reciprocal. For instance,6\times \dfrac{1}{6}=1, where \dfrac{1}{6} and 6 are inverses of one another. The concept of an inverse is often used when solving equations, like solving 5x=20.

\displaystyle 5x\displaystyle =\displaystyle 20Write down the equation
\displaystyle 5x\times \frac{1}{5}\displaystyle =\displaystyle 20\times \frac{1}{5}Multiply by the inverse of 5
\displaystyle x\displaystyle =\displaystyle 4Simplify both sides

The determinant of a 2 x 2 matrix

The inverse of 6 can also be written as 6^{-1}. As with this example, we have that: A^{-1} represents the inverse of matrix A.

The determinant is a number that is used to find the inverse of a matrix (if it exists). If the determinant is zero then the inverse is said to be undefined, this can be useful when solving systems of linear equations later.

Generally, the determinant of a matrix A is written as det(A) and is calculated as follows:

For a matrix A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} , the determinant is det(A)= \begin{vmatrix} a&b\\ c&d \end{vmatrix}=ad-bc .

Examples

Example 1

Evaluate the determinant \begin{vmatrix} -4&-6\\ 3&1 \end{vmatrix} .

Worked Solution
Create a strategy

Get the difference between the products of the diagonals.

Apply the idea
\displaystyle \text{Determinant}\displaystyle =\displaystyle \begin{vmatrix} -4&-6\\ 3&1 \end{vmatrix}
\displaystyle =\displaystyle -4\times 1-3\times \left(-6\right)Get the products of the diagonals.
\displaystyle =\displaystyle -4-\left(-18\right)Evaluate the products
\displaystyle =\displaystyle 14Evaluate the subtraction
Idea summary

The determinant of a matrix A is written as det(A) and is calculated as follows:

For a matrix A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} , the determinant is det(A)= \begin{vmatrix} a&b\\ c&d \end{vmatrix}=ad-bc .

The inverse of a 2 x 2 matrix

To find the inverse of a 2\times 2 matrix, say A, we swap the entries along the main-diagonal, and multiply the entries in the off-diagonal by -1. Then we multiply the result by \dfrac{1}{det(A)}.

For a matrix A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} , the inverse is A^{-1}=\dfrac{1}{det(A)} \begin{vmatrix} d&-b\\ -c&a \end{vmatrix} .

The inverse matrix A^{-1} has the property that when we multiply it by A, we get the identity matrix.AA^{-1}=A^{-1}A=I

Examples

Example 2

Does this matrix \begin{bmatrix} 4&2\\ -5&6 \end{bmatrix} have an inverse?

A
Yes
B
No
Worked Solution
Create a strategy

Check if the determinant of the matrix is not equal to 0.

Apply the idea
\displaystyle \text{Determinant}\displaystyle =\displaystyle \begin{vmatrix} 4&2\\ -5&6 \end{vmatrix}
\displaystyle =\displaystyle 4\times 6-2\times \left(-5\right)Get the products of the diagonals.
\displaystyle =\displaystyle 24+10Evaluate the products
\displaystyle =\displaystyle 34Evaluate the addition

Since the determinant is 34 \neq 0, then the matrix has an inverse. This means that the correct answer is Option A.

Example 3

Consider the matrix A= \begin{bmatrix} -8&2\\ -1&9 \end{bmatrix}.

a

Find the determinant of A.

Worked Solution
Create a strategy

Get the difference between the products of the diagonals of matrix A.

Apply the idea
\displaystyle \text{Determinant}\displaystyle =\displaystyle \begin{vmatrix} -8&2\\ -1&9 \end{vmatrix}
\displaystyle =\displaystyle -8\times9- \left(-1\right)\times 2Get the products of the diagonals.
\displaystyle =\displaystyle -72-(-2)Evaluate the products
\displaystyle =\displaystyle -70Evaluate the subtraction
b

Find the inverse A^{-1}.

Worked Solution
Create a strategy

Use the formula on the inverse of matrix given by A^{-1}=\dfrac{1}{det(A)} \begin{bmatrix} a_{22}&-a_{12}\\ -a_{21}&a_{11} \end{bmatrix} , where matrix A=\begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{bmatrix}.

Apply the idea

From part (a), the determinant is -70. So, the inverse of matrix A is given by:

\displaystyle A^{-1}\displaystyle =\displaystyle \dfrac{1}{-70}\begin{bmatrix} 9&-2\\ 1&-8 \end{bmatrix}Use the formula of inverse matrix
\displaystyle =\displaystyle \begin{bmatrix} 9 \times \dfrac{1}{-70}&-2 \times \dfrac{1}{-70}\\ \\ 1 \times \dfrac{1}{-70} &-8 \times \dfrac{1}{-70} \end{bmatrix}Multiply each element by \dfrac{1}{-70}
\displaystyle =\displaystyle \begin{bmatrix} \frac{-9}{70}&\frac{1}{35}\\ \\ \frac{-1}{70}&\frac{4}{35} \end{bmatrix}Evaluate

Example 4

Consider the matrix A= \begin{bmatrix} 3&1\\ 4&7 \end{bmatrix} and its inverse A^{-1}= \begin{bmatrix} \dfrac{7}{17}&-\dfrac{1}{17}\\ \\ n&\dfrac{3}{17} \end{bmatrix}. Solve for n.

Worked Solution
Create a strategy

Use the multiplicative inverse property of a matrix given by A \times A^{-1}=I to solve for n.

Apply the idea

Note that using the multiplicative inverse property for the given matrices, we have:

\displaystyle I\displaystyle =\displaystyle A \times A^{-1}
\displaystyle \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}\displaystyle =\displaystyle \begin{bmatrix} 3&1\\ 4&7 \end{bmatrix} \times \begin{bmatrix} \dfrac{7}{17}&-\dfrac{1}{17}\\ \\ n&\dfrac{3}{17} \end{bmatrix}

To solve for n, we multiply the elements on the first row of A by the elements on the first column of A^{-1} and equate the product to 1.

\displaystyle 1\displaystyle =\displaystyle 3 \times \left(\dfrac{7}{17}\right)+1\times nMultiply the corresponding elements
\displaystyle 1\displaystyle =\displaystyle \left(\dfrac{21}{17}\right)+ nEvaluate the multiplication
\displaystyle 1-\dfrac{21}{17}\displaystyle =\displaystyle \left(\dfrac{21}{17}\right)+ n-\dfrac{21}{17}Subtract \dfrac{21}{17} on both sides
\displaystyle n\displaystyle =\displaystyle \dfrac{-4}{17}Evaluate
Idea summary

The inverse matrix A^{-1} has the property that when we multiply it by A, we get the identity matrix.AA^{-1}=A^{-1}A=I

Outcomes

U1.AoS3.4

determinant and inverse of a matrix

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