Methods to Find Inverse of a Matrix
Methods to find the inverse of a matrix involve the inverse of a matrix formula and by elementary operations. The inverse of matrix A is represented as A-1 which when multiplied by matrix A gives an identity matrix.
In this article, we will explore different methods to find the inverse of a matrix in detail along with the inverse of matrix definition and inverse of matrix properties.
Table of Content
- What is Inverse of a Matrix?
- Inverse of a Matrix Definition
- Properties of Inverse of Matrix
- Methods to Find Inverse of a Matrix
- Inverse of a Matrix by Inverse of Matrix Formula
- Steps to Find Inverse of Matrix by Inverse of Matrix Formula
- Inverse of Matrix by Elementary Transformations
- Inverse of 2 Γ 2 Matrix
- Examples of Methods to Find Inverse of a Matrix
- Practice Problems on Methods to Find Inverse of a Matrix
What is Inverse of a Matrix?
Inverse of a Matrix is defined as the matrix when multiplied by the original matrix gives the identity matrix. If A is a matrix, then the inverse of matrix A is represented as A-1. The inverse of a matrix can only be determined for a square matrix and the determinant is not equal to zero (i.e., non-singular matrix).
Inverse of a Matrix Definition
The matrix results in identify matrix when multiplied by the given matrix A then the matrix is called as Inverse matrix of A. It is denoted as A-1.
AA-1 = A-1A = I
where,
- A-1 is Inverse of Matrix A
- I is an Identity Matrix
Properties of Inverse of Matrix
Some properties of the Inverse of the Matrix are given below.
- (A-1)-1 = A
- (AT)-1 = (A-1)T
- (AB)-1 = B-1 A-1
- (kA)-1 = k-1 A-1 = (1/k)A-1
Methods to Find Inverse of a Matrix
The different methods to find the inverse of a matrix are as follows.
- By Inverse of a Matrix Formula
- By Elementary Operations
Inverse of a Matrix by Inverse of Matrix Formula
The inverse of matrix formula is obtained by dividing adjoint of matrix by determinant of matrix. The adjoint of a matrix is the transpose of the cofactor matrix. The below formula represents the inverse of a matrix formula.
A-1 = adj(A) / |A|
where,
- A-1 is Inverse of Matrix A
- adj(A) is Adjoint of Matrix A
- |A| is Determinant of Matrix A
Steps to Find Inverse of Matrix by Inverse of Matrix Formula
The below steps are used to find inverse of matrix from the inverse matrix formula.
- Step 1: First, find the determinant of the matrix, if determinant is zero then, inverse of matrix does not exists and if the determinant is non-zero then follow further steps.
- Step 2: Find the adjoint matrix of the given matrix.
- Step 3: Then, divide the adjoint matrix by determinant of the matrix.
Inverse of Matrix by Elementary Transformations
The below steps are followed to find the inverse of a matrix using elementary transformations.
- Step 1: First, write the matrix as A = IA where I is identity matrix of same order as A.
- Step 2: Then perform row elementary operation or column elementary operation until we get identity matrix in LHS.
- Step 3: When identity matrix I is achieved by performing the row or column operation then we get I = BA
- Step 4: Then, B represents the inverse of matrix A.
Inverse of 2 Γ 2 Matrix
Inverse of 2 Γ 2 matrix A = [Tex]\begin {bmatrix} a & b \\ c & d \end{bmatrix}[/Tex] can be directly obtained by below formula.
A-1 = [Tex]\bold{\frac{1}{ad β bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}[/Tex]
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Examples of Methods to Find Inverse of a Matrix
Example 1: Find the inverse of the matrix P = [Tex]\begin {bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix}[/Tex] by Direct Method.
Solution:
We can find the inverse of matrix P by following formula
P-1 = [Tex]\bold{\frac{1}{ad β bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}[/Tex]
P-1 = [Tex]\frac{1}{(2\times1 )- (3\times 5)}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]
P-1 = [Tex]\frac{1}{2- 15}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]
P-1 = [Tex]\frac{1}{- 13}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]
Example 2: Find the inverse of matrix Q = [Tex]\begin {bmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{bmatrix}[/Tex] using inverse matrix formula.
Solution:
We can find the inverse of matrix Q by following formula.
Q-1 = adj(Q) / |Q|
First, we find |Q|.
|Q| =[Tex] \begin {vmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{vmatrix} [/Tex]
|Q| = 1[Tex] \begin {vmatrix} 1 & 0\\ 2&3 \end{vmatrix} [/Tex] β 0 [Tex] \begin {vmatrix} 6 & 0\\ 5&3 \end{vmatrix} [/Tex]+ 4 [Tex] \begin {vmatrix} 6& 1 \\ 5&2 \end{vmatrix} [/Tex]
|Q| = 1[3 β 0] β 0 + 4[12 β 5]
|Q| = 1(3) + 4(7)
|Q| = 3 + 28
|Q| = 31
Now we will find adj(Q)
To find adj(Q) we find the cofactor matrix of Q as
adj Q = Transpose of cofactor matrix of Q
Cij = (-1)i + j Mij
where, Mij is minor.
Cofactor(Q) = [Tex] \begin {bmatrix}(-1)^{1+1}\begin {vmatrix} 1 & 0\\ 2&3\\ \end{vmatrix} (-1)^{1+2}\begin {vmatrix} 6 & 0\\ 5&3\\ \end{vmatrix} (-1)^{1+3}\begin {vmatrix} 1 & 0\\ 2&3\\ \end{vmatrix} \\ \\ (-1)^{2+1}\begin {vmatrix} 0 & 4\\ 2&3\\ \end{vmatrix} (-1)^{2+2}\begin {vmatrix} 1 & 4\\ 5&3\\ \end{vmatrix} (-1)^{2+3}\begin {vmatrix} 1 & 0\\ 5&2\\ \end{vmatrix} \\ \\ (-1)^{3+1}\begin {vmatrix} 0 & 4\\ 1&0\\ \end{vmatrix} (-1)^{3+2}\begin {vmatrix} 1 & 4\\ 6&0\\ \end{vmatrix} (-1)^{3+3}\begin {vmatrix} 1 & 0\\ 6&1\\ \end{vmatrix} \\ \\ \end {bmatrix}[/Tex]
On solving above matrix we get
Cofactor of Q = [Tex]\begin {bmatrix} 3 & -18 & 7 \\ 8& -17 & -2\\ -4&24&1 \end{bmatrix} [/Tex]
adj(Q) = [Cofactor(Q)] T
adj(Q) = [Tex]\begin {bmatrix} 3 & -18 & 7 \\ 8& -17 & -2\\ -4&24&1 \end{bmatrix}^T[/Tex]
adj(Q) = [Tex]\begin {bmatrix} 3 & 8 & -4 \\ -1 8& -17 & 24\\ 7&-2&1 \end{bmatrix} [/Tex]
So, the inverse of matrix Q is given by
Q-1 = (1 / 31) [Tex]\begin {bmatrix} 3 & 8 & -4 \\ -1 8& -17 & 24\\ 7&-2&1 \end{bmatrix} [/Tex]
Example 3: Find the inverse of matrix V = [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5 &0 & 2 \end{bmatrix}[/Tex] by elementary transformations.
Solution:
To find the inverse of the matrix V = [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] we will use row operation.
V = IV
[Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex]
R1 β R1 β R3
[Tex]\begin {bmatrix} 1 & 2 & 1 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] V
R3 β R3 β 5R1
[Tex]\begin {bmatrix} 1 & 2 & 1 \\ 0 & 1 & 4\\ 4&-10 &-3 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0\\ -5&0 & 6 \end{bmatrix}[/Tex] V
R1β R1 β 2R2
[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&-10 & -3 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5&0 & 6 \end{bmatrix}[/Tex] V
R3 β R3 + 10R1
[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&0 & 37 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5&10 & 6 \end{bmatrix}[/Tex] V
R3 β R3/37
[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V
R1 β R1 + 7R3
[Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 4\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 0 & 1 & 0\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V
R2 β R2 β 4R3
[Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 9/37 & -3/37 & -24/37\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V
Since, the above expression is of form I = BV
Inverse of matrix V = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 9/37 & -3/37 & -24/37\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex]
Practice Problems on Methods to Find Inverse of a Matrix
P1: Find the inverse of the matrix P = [Tex]\begin {bmatrix} 12 & 8 \\ 20 & 15 \end{bmatrix}[/Tex] by Direct Method.
P2: Find the inverse of matrix X = [Tex]\begin {bmatrix} 2 & 10 \\ 15 & 5 \end{bmatrix}[/Tex] by elementary transformations.
P3: Find the inverse of matrix B = [Tex]\begin {bmatrix} 3 & 1 & -1 \\ 2& -2 & 0\\ 1&2&-1 \end{bmatrix}[/Tex] by inverse matrix formula.
P4: Find the inverse of matrix D = [Tex]\begin {bmatrix} 2 & 3 & 1 \\ 1 & 1 & 2\\ 2&3&4 \end{bmatrix}[/Tex] by elementary transformations.
Methods to Find Inverse of a Matrix β FAQs
What is Inverse of a Matrix Formula?
Inverse of a matrix formula is given by:
A-1 = adj(A) / |A|
How to Find an Inverse Matrix?
To find an inverse matrix we divide the adjoint matrix by determinant of matrix. Another way to find an inverse matrix is to perform elementary operations on it.
What are Different Ways to Find Inverse of a Matrix?
Different ways to find inverse of a matrix are:
- By Inverse of Matrix Formula
- By Elementary Transformations
What are the methods to find the inverse of a matrix?
There are several methods to find the inverse of a matrix:
- Gaussian elimination:
- Matrix adjoint method:
- Elementary row operations:
- Using determinants:
Which method is the most efficient for finding the inverse of a matrix?
The efficiency of each method depends on various factors such as the size of the matrix, computational resources available, and the specific properties of the matrix. In general, Gaussian elimination is often preferred for larger matrices due to its efficiency, while using the adjoint method might be more convenient for smaller matrices.
Are there any special types of matrices for which finding the inverse is easier?
Yes, certain types of matrices have properties that make finding their inverses easier. For example, diagonal matrices and triangular matrices have straightforward methods for finding their inverses. Additionally, if a matrix is orthogonal or unitary, its inverse is simply its transpose.
Can all matrices be inverted?
No, not all matrices can be inverted. Only square matrices that are non-singular, meaning they have a non-zero determinant, have inverses. If the determinant of a matrix is zero, it is called a singular matrix, and it does not have an inverse.
What is the importance of finding the inverse of a matrix?
Finding the inverse of a matrix is important in various mathematical and practical applications. It allows solving systems of linear equations, computing solutions to optimization problems, and performing transformations in areas like computer graphics and engineering.
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