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 – 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] – 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 – 0] – 0 + 4[12 – 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 ← R1 – 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 ← R3 – 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
R1← R1 – 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 ← 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 ← 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 ← 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 ← R2 – 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 – FAQs
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:
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.
