A matrix has an inverse if and only if it is a square matrix with the same number of rows and columns and its determinant is non-zero. Such a matrix is called non-singular or invertible.
No, only non-singular (invertible) square matrices can be inverted. A matrix is considered non-singular if its determinant is non-zero. Conversely, if the determinant is zero, the matrix is singular and does not have an inverse
The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix. It is used in the process of finding the inverse of a matrix.
To understand how to find the inverse of a matrix, it's essential to first grasp what a matrix is. A matrix is a structured rectangular array of numbers, known as elements. These elements are organized into horizontal rows and vertical columns. The dimensions of a matrix are denoted by its number of rows and columns. If a matrix has ‘a’ rows and ‘b’ columns, its order is expressed as a × b, where ‘a’ and ‘b’ are positive integers.
For a given matrix A of order, a × b, the inverse of matrix A, if it exists, is denoted as A−1. The inverse of a matrix is a crucial concept in linear algebra and is defined such that when matrix A is multiplied by its inverse A−1, the result is the identity matrix III, which is a square matrix with ones on the diagonal and zeros elsewhere. Mathematically, this is represented as:
A⋅ A−1 = A−1⋅A = I
A matrix has an inverse only if it is a square matrix, meaning the number of rows is equal to the number of columns and its determinant is non-zero.
1.0Inverse of a Matrix Definition
If A is a non-singular (invertible) square matrix, then there exists an inverse matrix denoted by A–1. This inverse matrix satisfies the condition:
AA−1=A−1A=I
where I is the identity matrix.
2.0How to Find the Inverse of a 3 × 3 Matrix:
To calculate the inverse of a matrix, follow these steps:
Find the Matrix of Minors: Calculate the minor for each element of the original matrix.
Convert to the Matrix of Cofactors: Apply the checkerboard pattern of signs to the matrix of minors.
Find the Adjoint (Adjugate) of the Matrix: Take the transpose of the matrix of cofactors.
Multiply by det(A)1: Compute the determinant of the original matrix and multiply the adjoint matrix by the reciprocal of the determinant.
This process will yield the inverse of the original 3 × 3 matrix.
A=adgbehcfi
Solution: First of all we will calculate the matrix of Minors.
To find the inverse of a 3 × 3 matrix, let's go through the process in detail.
Step 1: Calculate the Matrix of Minors
Given a 3 × 3 matrix A:
A=adgbehcfi
The matrix of minors is obtained by calculating the minor for each element. The minor of an element is the determinant of the 2 × 2 submatrix formed by deleting the row and column of that element.
Next, we apply the checkerboard pattern of signs to the matrix of minors to obtain the matrix of cofactors. The signs alternate starting with a positive sign in the top-left corner.
Finally, we need to calculate the determinant of \(A\):
det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)
Then, the inverse of A is:
A−1=det(A)1Adj(A)
By following these steps, we can find the inverse of any 3 × 3 matrix, provided that the determinant is non-zero.
3.0Example on How to Find Inverse of Matrix
Example: Find the Inverse of a 3 × 3 Matrix
Let us calculate the inverse of the following 3 × 3 matrix A:
A=105216340
Step 1: Calculate the Matrix of Minors
Minor of a11 = 1:
M11=1640=(1⋅0−4⋅6)=−24
Minor of a12 = 2:
M12=0540=(0⋅0−4⋅5)=−20
Minor of a13 = 3:
M13=0516=(0⋅6−1⋅5)=−5
Minor of a21 = 0:
M21=2630=(2⋅0−3⋅6)=−18
Minor of a22 = 1:
M22=1530=(1⋅0−3⋅5)=−15
Minor of a23 = 4:
M23=1526=(1⋅6−2⋅5)=−4
Minor of a31 = 5:
M31=2134=(2⋅4−3⋅1)=5
Minor of a32 = 6:
M32=1034=(1⋅4−3⋅0)=4
Minor of a33 = 0:
M33=1021=(1⋅1−2⋅0)=1
The matrix of minors is:
Minors(A)=−24−185−20−154−5−41
Step 2: Convert to the Matrix of Cofactors
Apply the checkerboard pattern of signs:
Cofactors(A)=−2418520−15−4−541
Step 3: Find the Adjoint (Adjugate) of the Matrix
The adjoint is the transpose of the matrix of cofactors:
Adj(A)=−2420−518−1545−41
Step 4: Calculate the Determinant
The determinant of A is:
det(A)=1(1⋅0−4⋅6)−2(0⋅0−4⋅5)+3(0⋅6−1⋅5)
det(A)=1(0−24)−2(0−20)+3(0−5)
det(A)=−24+40−15
det(A)=1
Step 5: Multiply by det(A)1
Since det(A)=1, the inverse is simply the adjoint matrix:
A−1=11⋅Adj(A)
A−1=−2420−518−1545−41
Thus, the inverse of matrix A is:
A−1=−2420−518−1545−41
Here are some practice questions to help you master finding the inverse of a 3 × 3 matrix:
4.0Practice Questions on How to Find Inverse of Matrix
Matrix A: A=2133041−12
Matrix B:B=432765213
Matrix C:C=105216340
Matrix D: D=1−13034210
Matrix E:E=232121112
Answers:
A−1=−3435−3432−31−31−1−11
B−1=3−31−1710−211−61
C−1=−2420−518−1545−41
D−1=−41520−310−1
E−1=3−2−1−120−101
5.0Sample Questions on Find Inverse of Matrix
1. What is the inverse of a matrix?
Ans: The inverse of a matrix A is another matrix, denoted as A−1 , such that when A is multiplied by A–1, the result is the identity matrix I. Mathematically, AA–1 = A–1A = I, where I is called the identity matrix.
2. When does a matrix have an inverse?
Ans: A matrix has an inverse if and only if it is a square matrix with the same number of rows and columns and its determinant is non-zero. Such a matrix is called non-singular or invertible.
3. How do you find the inverse of a 2 × 2 matrix?
Ans: For a 2 × 2 matrix A=(acbd) the inverse is given by A−1=ad−bc1(d−c−ba), provided .
4. What are the steps to find the inverse of a 3 × 3 matrix?
Ans: To find the inverse of a 3 × 3 matrix, follow these steps:
Calculate the matrix of minors.
Convert the matrix of minors to the matrix of cofactors.
Find the adjoint (transpose of the matrix of cofactors).
Calculate the determinant of the original matrix.
Multiply the adjoint by det(A)1.
5. Can every square matrix be inverted?
Ans: No, only non-singular (invertible) square matrices can be inverted. A matrix is considered non-singular if its determinant is non-zero. Conversely, if the determinant is zero, the matrix is singular and does not have an inverse.
6. What is the identity matrix?
Ans: The identity matrix, denoted as I, is a square matrix with ones on the main diagonal and zeros elsewhere. For a 3 × 3 matrix, it looks like: I=100010001
Multiplying any matrix by the identity matrix leaves the original matrix unchanged.
Table of Contents
1.0Inverse of a Matrix Definition
2.0How to Find the Inverse of a 3 × 3 Matrix:
3.0Example on How to Find Inverse of Matrix
4.0Practice Questions on How to Find Inverse of Matrix