Determinants

Calculation of the adjoint, inverse of a matrix is one of the key topics in algebra that is often part of the syllabus of various competitive and academic examinations. There are three methods to calculate the inverse of a matrix – using elementary row operation, using a scientific calculator or simply using the adjoint of the matrix. We will focus on the latter in this article. An important thing to be noted here is that not all matrices have an inverse. In order to check if the matrix in question has a possible inverse, calculate the determinant of the matrix, and if its value is equal to 0, it doesn’t have an inverse. 

Adjoint of a Matrix

Before defining the adjoint of a matrix, one needs to be familiar with a few terms. These can also be seen as the steps in calculating the adjoint.

Minors 

Minors are related to the elements of the matrix. These are calculated for each element and can be found by deleting the row and column where the element resides. This can be explained with the help of the following example –

is a matrix of order 2. Here, element 3 is positioned in the 1st column of the 1st row. Hence, we will delete 2, which resides in the same row, and we also delete 1, which resides in the same column. Thus, the remaining element, i.e. 5, is its minor. 

Co-factors

Cofactors of an element is given by the formula Aij = (-1) (i+j) Mij

Let’s use the same example to find the cofactor of element 3. We have obtained the value of the minor as 5 for the element 3. We are aware that the ith and jth characters denote the position of the row and column of the element, respectively. So for element 3, the value of i is 1, and the value of j is also 1. Hence, inserting this in the formula, we find –

A11 = (-1) (1+1) = 1 5 = 5

Similarly, cofactors for the other elements can be obtained, and the final matrix A in this case

is  . 

Transpose of a Matrix

The transpose of a matrix is denoted by AT. In simple terms, it is turning over the matrix. It can be done by interchanging the rows with the columns and vice versa. 

Taking the above example of matrix A= , transpose the matrix A = AT =  .

Defining Adjoint of a matrix

Now, taking the above terms into consideration, we can define the adjoint of a matrix. 

The adjoint of a matrix of order n is simply the transpose of the matrix’s cofactors. It’s denoted by the term adj A for a matrix A. It’s also known as the adjugate matrix.

  Adj A = [Aij]nn  

It is simply obtained by first finding the minors, then the cofactors and finally transpose of the matrix as explained above. In this case, the final matrix A obtained at last is the adjoint of matrix A. 

Inverse of a Matrix

The inverse of a matrix is defined as the product of its adjoint divided by the matrix’s determinant. In simple terms, a matrix A’s inverse is another matrix B as such that if A and B are multiplied, then the product is the multiplicative identity.

It can be expressed using the following equation AB = BA = I

The formula for inverse of a matrix A is given by 

A-1  = 1|A| (Adj A)

One important thing to be noted is that a square matrix is only invertible if it is non-singular in nature. It means that the value of the determinant of the matrix must not be zero. The reason behind this is if the value of the determinant is zero, then inserting the same in the formula gives the overall value as 0. But we know that the product of the matrix and its inverse must be equivalent to 1. Hence, none of the values can be 0. 

Adjoint, inverse of square matrix ( 22 )

This is a sample problem that will explain step-by-step the calculation of inverse in case of a matrix of order 2. We will take the Matrix A, as discussed earlier. 

Step 1. Find the determinant of the matrix A= .. 

|A| = (35) – (21) = 13

Step 2. Find the adjoint of the matrix A.

We have already calculated the adjoint of matrix A as

Step 3. Calculate the inverse.

A-1  = 1|A| (Adj A)

=

=

You can cross-check your answer to this by multiplying the inverse with the original matrix. If the answer comes to be 1 then your inverse is correct. 

Adjoint, inverse of square matrix (33 )

Following is an example of calculating the inverse of an order 3 matrix. We will use Matrix Z for the purpose. 

Z =

Step 1. Find the determinant of the matrix. 

|Z| = 1(20 – 1-4) – 6(50-8) + 8(51+2-2)

     = 60 

Since it is established that matrix Z is non-singular, we can proceed with the calculations.

Step 2. Finding the adjoint. 

• The first step in finding the adjoint is identifying the minors. For a 33 matrix, we need to delete the row and column of where the element is positioned. Then find the difference of product of the diagonal elements. 

 

• Using this, the minors obtained will give the following matrix –

The next step is multiplying the matrix obtained with a sign chart. The sign chart works on the same cofactor formula discussed earlier. 

 

Transpose the resulting matrix to obtain Adj Z. 

 Adj Z = Transpose of  = 

Step 3. Finding the inverse

Z-1  = 1|A| (Adj Z)

=

=

Conclusion

The calculation of adjoint, Inverse of square matrix only works for a non-singular matrix. In order to find the inverse of a matrix A, first calculate the minors, cofactors and transpose the matrix. The resulting matrix is the adjoint of A, and this value can be applied to the inverse formula to find A-1. The answer can be tested by multiplying the inverse with the original matrix, and if the value is 1, then the inverse is correct.