How to Find the Inverse of a Matrix?

Finding the inverse of a matrix is as simple as finding the additive or multiplicative inverse of any number. The inverse of a matrix can be found using different techniques like elementary-row operation, adjoint method and determinants. This article will discuss these methods in complete detail with the help of examples.

Matrix inversion is a critical concept in mathematics, particularly in areas where systems of linear equations are involved. Think of it as finding a key to unlock a coded message. Just as multiplying a number by its inverse (or reciprocal) gives you 1, multiplying a matrix by its inverse results in the identity matrix, which acts like the number 1 in matrix algebra.
Example: If A and B are two matrices, then B is said to be the inverse of A if and only if A*B = B*A = I, where I is an Identity Matrix.
If B satisfies the above condition, then B = A^-1

Let's take a real-life example to understand the Inverse of a Matrix better.

Consider a scenario where you need to solve for x and y in the following system of equations:

1. 3x + 4y = 10
2. 5x + 6y = 12

This system can be represented as a matrix equation, A*X = B, where:

• A is the matrix of coefficients

• X is the matrix of variables (x, y).
• B is the matrix of constants (10, 12).

Check out the video to get a better understanding of how to find the inverse of a matrix

Methods to Find Inverse of a Matrix

• Elementary Row Operation
• Adjoint Methods

Let's discuss them one by one in complete detail.

Elementary Row Operation

This method uses different elementary row operations to transform a matrix into an identity matrix.

Here, we mainly perform three elementary row operations:

Row Switching: Swap two rows

Row Multiplication: Multiplying all entries in a row by a non-zero constant.

Row Addition: Add a multiple of one row to another row.

Now, let's take an example to get a better understanding.

Example-1: Find the inverse of the given matrix.

In order to find the inverse of the given matrix, we will transform the given matrix (A) to the identity matrix. To do that, we will use an augmented matrix that looks like this:

Step-1: Multiply the first row by (1/2) to make the a₁₁ = 1.
i.e., R₁ -> (1/2) R₁

Step-2: Subtract 3 times the first from the second row to make the a₂₁ = 0. 
i.e., R₂ -> R₂ - 3R₁

Step-3: Divide the second row by 3 to make a₂₂ = 1.
i.e., R₂ -> (1/3)R₂

Step-4: Finally, subtract 2 times the second row from the first row to make a₁₂ = 0.
i.e., R₁ -> R₁-2R₂

Hence, the inverse of the matrix A is:

From above, you must understand how to find the inverse of a matrix using the elementary row operation.

Now, here is a question for your practice.

Find the inverse of the given matrix using elementary row operation.

Let's move to the second method to find the inverse of a matrix.

Adjoint Method

To find the inverse of a matrix using the adjoint method, we must know the concepts of Minor and Cofactor.
Minor: The minor of an element is the determinant of the smaller matrix obtained by removing the row and column of the element.
Cofactor: The cofactor of an element is the multiplication of (-1)^(i+j) with the minor of that element. 
Steps to Find the Inverse of a Matrix using the Adjoint Matrix

1. Find the cofactor.
2. Transpose the cofactor matrix.
3. Calculate the determinant.
4. Divide adjoint by determinant.
Let's take an example to get a better understanding.

Step-1: Find the cofactor matrix

• Cofactor of 4 (a₁₁): (−1)¹⁺¹×det⁡([1]) = 1(−1)1+1×det([1]) = 1
• Cofactor of 3 (a₁₂): (−1)¹⁺²×det⁡([2]) = −2(−1)1+2×det([2]) = −2
• Cofactor of 2 (a₂₁): (−1)²⁺¹×det⁡([1]) = −1(−1)2+1×det([1]) = −1
• Cofactor of 1 (a₂₂): (−1)²⁺²×det⁡([4]) = 4(−1)2+2×det([4]) = 4

Hence, the corresponding cofactor matrix is:

Step-2: Find the adjoint matrix

The adjoint of a matrix is the transpose of the cofactor.

Step-3: Calculate the Determinant of A

det(A) = 4 × 1 − 3 × 2 = −2

Step-4: Divide the Adjoint by the Determinant

Hence, from the above calculation, we can conclude that if it exists, it is given by A^-1 = 1/det(A) x adj(A)

where,

det(A): determinant of A

adj(A): Adjoint of A

Common Mistakes To Avoid While Finding the Determinant

• Remember to check if the determinant of a matrix is zero since a matrix is invertible if and only if the determinant is non-zero.
• While using elementary row operation, it is very common to make calculation errors. SO double-check the calculation.
• While finding the adjoint, 
• For a matrix greater than 2x2, it is complex to calculate the cofactor. So, correctly identify the submatrix and calculate the determinant.
• It is important to remember that cofactor involves multiplication (-1)^(i+j) in the minor.
• After finding the cofactor, remember to transpose the matrix to get the adjoint matrix.

Conclusion

In this article, we have briefly discussed different methods to find the inverse of a matrix, i.e., row elimination method and adjoint method. Hope you will like the article.

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