Inverse Matrices

Do you know that just as real numbers other than zero can have an inverse, matrices can have inverses too? Hereafter, you would understand how to calculate the inverse of matrices.

Get started Sign up for free
Contents
Table of contents

    Definition of Inverse matrices

    A matrix is said to be the inverse of another matrix if the product of both matrices results in an identity matrix. However, before going into inverse matrices we need to refresh our knowledge of identity matrix.

    What is an Identity matrix?

    An identity matrix is a square matrix in which when multiplied by another square matrix equals to the same matrix. In this matrix, the elements from the topmost left diagonal to the downmost right diagonal is 1 while every other element in the matrix is 0. Below are examples of a 2 by 2 and 3 by 3 identity matrix respectively:

    A 2 by 2 identity matrix:

    1001

    A 3 by 3 identity matrix:

    100010001

    Thus, the inverse of a matrix can be derived as:

    Where I is the identity matrix and A is a square matrix, then:

    A×I=I×A=A

    To have a little insight on this, consider:

    A×I=AI=A×A-1

    A-1 is the inverse of matrix A. The equation:

    I=A×A-1

    means that the product of matrix A and inverse matrix A would give I, the identity matrix.

    Therefore, we can verify if two matrices being multiplied are inverse of each other.

    Verify if the following are inverse matrices or not.

    a.

    A=22-14 and B=1212-114

    b.

    M=3412 and N=1-2-1232

    Solution:

    a. find the product between matrix A and B;

    A×B=22-14×1212-114A×B=(2×12)+(2×(-1))(2×12)+(2×14)(-1×12)+(4×(-1))(-1×12)+(4×14)A×B=1-21+12-12-4-12+1A×B=-1112-41212

    Since the product of matrix A and B fails to give an identity matrix, hence, A is not an inverse of B and vice versa.

    b.

    M×N=3412×1-2-1232M×N=(3×1)+(4×(-12))(3×(-2))+(4×32)(1×1)+(2×(-12)(1×(-2))+(2×32)M×N=3-2-6+61-1-2+3M×N=1001

    Since the product of matrices M and N yields an identity matrix, it means matrix M is the inverse of matrix N.

    What methods are used in finding the inverse of matrices?

    There are three ways of finding the inverse of matrices, namely:

    1. Determinant method for 2 by 2 matrices.

    2. Gaussian method or augmented matrix.

    3. The adjoint method through the use of matrix cofactors.

    However, at this level, we shall only learn the determinant method.

    Determinant method

    In order to find the inverse of a 2 by 2 matrix, you should apply this formula:

    M=abcdM-1=1ad-bcd-b-ca

    Provided that:

    ad-bc0

    Where the determinant of a matrix is 0, there is no inverse.

    Therefore, the inverse of a 2 by 2 matrix is the product of the inverse of the determinant and the matrix being altered. The altered matrix is gotten by swapping the diagonal elements with the cofactor sign on each.

    Find the inverse of matrix B.

    B=1023

    Solution:

    B=1023

    Using;

    abcd-1=1ad-bcd-b-ca

    Then;

    B-1=1(1×3)-(0×2)30-21B-1=13-030-21B-1=1330-21

    or,

    B-1=1330-21 =330-2313 B-1= 10-2313

    Most importantly, once your determinant is calculated and your answer is equal to 0, it just means that the matrix has no inverse.

    The inverse of 3 by 3 matrices can also be derived using:

    M-1=1Madj(M)

    Where,

    Mis the determinant of a matrix M

    adj(M) is the adjoint of matrix M

    To achieve this, four basic steps are followed:

    Step 1 - Find the determinant of the given matrix. If the determinant is equal to 0, it means no inverse.

    Step 2 - Find the cofactor of the matrix.

    Step 3 - Transpose of the cofactor matrix to give the adjoint of the matrix.

    Step 4 - Divide the adjoint matrix by the determinant of the matrix.

    Examples of inverse matrices

    Let's have some more examples to understand inverse matrices better.

    Find the inverse of the matrix X.

    X=21-3530-421

    Solution:

    This is a 3 by 3 matrix.

    Step1: Find the determinant of the given matrix.

    X=23021-150-41-353-42X=2(3-0)-1(5-0)-3(10+12)X=6-5-66X=-65

    Since the determinant is not equal to 0, it means that the matrix X has an inverse.

    Step2: Find the cofactor of the matrix.

    The cofactor is calculated with

    Cij=(-1)i+j×Mij

    The cofactor of 2 which is C11 is

    C11=(-1)1+1×3021 C11=1(3-0)C11=3

    The cofactor of 1 which is C12 is

    C12=(-1)1+2×50-41 C12=-1(5-0)C12=-5

    The cofactor of -3 which is C13 is

    C13=(-1)1+3×53-42 C13=1(10+12)C13=22

    The cofactor of 5 which is C21 is

    C21=(-1)2+1×1-321 C21=-1(1+6)C21=-7

    The cofactor of 3 which is C22 is

    C22=(-1)2+2×2-3-41 C22=1(2+12)C22=14

    The cofactor of 0 which is C23 is

    C23=(-1)2+3×21-42 C23=-1(4+4)C23=-8

    The cofactor of -4 which is C31 is

    C31=(-1)3+1×1-330 C31=1(0+9)C31=9

    The cofactor of 2 which is C32 is

    C32=(-1)3+2×2-350 C32=-1(0+15)C32=-15

    The cofactor of 1 which is C33 is

    C33=(-1)3+3×2153 C33=1(6-5)C33=1

    So the cofactor of the matrix X is

    Xc=3-522-714-89-151

    Step 3: Transpose of the cofactor matrix to give the adjoint of the matrix.

    the transpose of Xc is

    (Xc)T=Adj(X)=3-79-514-1522-81

    Step 4: Divide the adjoint matrix by the determinant of the matrix.

    Remember the determinant of matrix X is 65. This final stage gives us the inverse of matrix X which is X-1. Hence, we have

    X-1=1-653-79-514-1522-81X-1=-365765-965565-14651565-2265865-165X-1=[-365765-965113-1465313-2265865-165]

    Using matrix operations solve for x and y in the following:

    2x+3y=6x-2y=-2

    Solution:

    This equation can be represented in matrix form as

    231-2xy=6-2

    Let the matrices be represented by P, Q and R respectively such that

    P×Q=R

    We intend on finding matrix Q since it represents our unknowns x and y. So we make matrix Q the subject of the formula

    P-1×P×Q=P-1×RP-1×P=I

    I is an Identity matrix and its determinant is 1.

    IQ=R×P-1Q=R×P-1

    P-1=231-2-1P-1=1(-4-3)-2-3-12P-1=273717-27

    Then,

    Q=273717-27×6-2Q=(27×6)+(37×-2)(17×6)+((-27)×-2)Q=127-6767+47Q=67107xy=67107x=67y=107

    Inverse Matrices - Key takeaways

    • A matrix is said to be the inverse of another matrix if the product of both matrices results in an identity matrix.
    • Inverse of a matrix is possible for a square matrix where the determinant is not equal to 0.
    • The inverse of a two-by-two matrix is obtained using: abcd-1=1ad-bcd-b-ca
    Frequently Asked Questions about Inverse Matrices

    How do you inverse the sum of two matrices?

    You can calculate the inverse of the sum of two matrices by adding the two matrices, then applying the formula for inverse matrices on it.

    What are the examples of matrices that can have an inverse?

    Any matrix that has its determinant not equal to 0 is an example of a matrix that has an inverse.

    How do you do the inverse of a  3x3 matrix?

    To get the inverse of a 3 by 3 matrix, you need to find the determinant first. Then, divide the adjoint of the matrix by the determinant of the matrix.

    How do you get the inverse of matrices in multiplication?

    To get the inverse of matrices in multiplication, find the product of the matrices. Then, use the formula on the new matrix to find its inverse.

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 6 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App
    Sign up with Email

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner