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Determinant of Inverse Matrix

Determinant of Inverse Matrix

We can write a given set of linear equations

as a single matrix equation

and this reformulation makes it easy to solve. So essentially to get x and y we divide the whole equation with the matrix on the left-hand side of the equality.

But what does it mean to take the inverse of a matrix and are we allowed to do it?

Also, we know if we want to divide by a number then we require the number to be nonzero.

Analogously, we have a necessary and sufficient condition for finding the inverse of a matrix through a unique real number associated with certain matrices called the determinant. These are essentially the concepts discussed in this article.

Determinant of a matrix

To every square matrix A of order n, we can associate a unique real number called the determinant of matrix A. It is denoted by or det A and is read as "determinant of A".

If .

The determinant is sometimes denoted by .

Note that only square matrices have a determinant.

How to find the value of the determinant of a square matrix of order 2?

As I said, a determinant can be found only for square matrices. We will see for each order now to find the determinant.

For a square matrix of order 1, the determinant is nothing but the element itself.

For a square matrix of order 2, say , we have.

The value of this determinant is found by subtracting the multiplication of the elements in the off-diagonal from the multiplication of the elements in the main diagonal.

Consider the matrix . We see that A is a square matrix and the order is 2.

We can find the determinant of A by subtracting the multiplication of the elements in the off-diagonal from the multiplication of the elements in the main diagonal.

The main diagonal entries of this matrix are 1 and 5. The off-diagonal entries of the matrix are 3 and 4.

Note that the value of a determinant can be a negative number as in the case above.

Can you find the determinant of a row matrix?

The answer is no unless it has only one row. Since, in a general row matrix, there is only one column but there can be any number of rows.

Examples of row matrices are and .

So unless it has only one row it is not a square matrix. We know that we can find determinants for only square matrices.

Minors and Cofactors of the elements of the determinant

For square matrices of order 3, say , we have

The value of this determinant is found by first finding the cofactors of the elements of this determinant. We will see in the next section how to find the cofactors of the elements in a determinant and how we could use it to find the determinant of a square matrix of order 3.

Let , be a determinant of order 3.

The determinant of order 2 obtained by deleting the corresponding rows and columns of a particular element of the determinant is called the minor of the element of the determinant.

It is denoted by Mij if the element is in the ith row and jth column of the determinant.

For the above determinant with 9 elements, we have 9 minors corresponding to each element. For element a in the first row, and first column of the determinant the minor is the determinant of order 2 by deleting the first row and first column. We get, . Similarly, we can find the minors of the other elements in a similar way;

The number (–1)i+j Mij is called the cofactor of the element.

The cofactor of the element in the ith row and jth column is usually denoted by Aij.

The cofactors of the elements of the determinant in the first row are given by

Given a matrix , we can write the determinant of this matrix .

We wish to calculate the cofactors of its elements 6, 3 and 7.

First, we see that 6 is in the 1st row and 3rd column. We can calculate the minor by deleting the 1st row and 3rd column from the determinant.

We get . Now we can obtain the cofactor of 6 using the formula

Similarly we can get the cofactors A22 and A31 corresponding to the elements 3 and 7.

How to find the value of determinant of a square matrix of order 3 or higher?

The value of the determinant of a square matrix of order 3 or higher is the sum of the products of elements of any row or column of a determinant with its corresponding cofactors.

Now we formulate the steps involved in finding the value of the determinant of a matrix of order n > 2.

  1. Step 1: Choose a row or column of the determinant you want to use to do the calculations for finding the value of the determinant. (It is irrelevant on your choice the answer would be the same)

  2. Step 2: Obtain the cofactor of all the elements in the chosen row or column.

  3. Step 3: The value of the determinant is then obtained using the above definition.

For the square matrix of order 3, say , we saw how to calculate the cofactors of the elements of its determinant .

To find the value of the determinant we only need to select the cofactors of one row or column. Let us choose the row one. The value of the cofactors of the elements a, b and c in the first row is given by

.

Now we can say the value of the determinant det A is given by the sum of the products of elements with its corresponding cofactors

,

which is equivalent to

.

Consider the square matrix of order 3, . Its determinant is .

Step 1: To calculate its determinant we select either a row or column and find the cofactors of these elements. Let us select the first row.

Step 2: The cofactors of the three elements 1, 2 and 6 are given as below

Step 3: Now that we have the cofactors we can find the value of the determinant by the sum of the products of the elements with the corresponding cofactors.

.

Inverse of a matrix

We are familiar of how to calculate the inverse of a number. What is the multiplicative inverse of 2? It is nothing but . That is if I multiply these two numbers 2 and , I will get the identity 1. Similarly the multiplicative inverse of 3 is .

Now we are interested in the same concept for matrices. What is the multiplicative inverse of a matrix? If I have a matrix A, then what is A-1 ?

Let A be a square matrix of order n, then we say the matrix A is invertible, in other words A-1 exists, if and only if the . In this case, the formula to obtain the inverse of the matrix A is given by

.

Here, adj A denotes a matrix called the adjoint of A. (We will see going forward how to obtain adjoint matrix)

By the above definition we see that if one wishes to find the inverse of a matrix then the first step is to check whether the determinant of the matrix is non-zero. Now to understand the formula for the inverse of a matrix we first have to know what is an adjoint of a matrix and how to perform the operation of transpose on matrices.

Transpose of a matrix

The transpose of a matrix A of order is obtained by interchanging the rows and columns of a matrix. It is denoted by AT. The order of AT will be .

Suppose you have the matrices .

The order of the matrices A and B are and respectively. The order of the transpose matrices AT and BT will be and respectively.

First let us write the transpose of A. The first column of the matrix AT is the first row of the matrix A. That is,

.

Similarly we write the second and third columns of the matrix AT from the second and third rows of the matrix A and we have

Similarly we can find the matrix BT of order to be

Adjoint of a matrix

Adjoint of a square matrix is the transpose of the Cofactor matrix.

The adjoint of a square matrix A is denoted by adj A.

A cofactor matrix is formed by the cofactor of the elements of the determinant of the matrix. The cofactor matrix of a square matrix A is denoted by cof A.

The following example will help you understand the above definition more clearly.

Let us consider a square matrix of order 3, say . The determinant of A is .

From the previous section we know how to calculate the cofactor of each element of this determinant.

The cofactor matrix is formed with the above as its elements.

Now we obtain the adjoint of the matrix A by taking the transpose of this cofactor matrix.

Given the matrix , we want to calculate its adjoint matrix.

Solution

For the same we first formulate the cofactor matrix of A which is obtained by finding the cofactors of all the elements of the matrix A.

From the above values we get the cofactor matrix of A as

We now can obtain the adjoint by taking the transpose of the above cofactor matrix

.

Finding the inverse of a matrix

Now that we have understood what an adjoint matrix is and how to take determinant of a matrix, we are all set to apply the formula for finding the inverse of a matrix. Lets write down the steps involved in finding the inverse of a matrix A.

  1. Step 1: Find the determinant of the matrix A and check whether . If the determinant is nonzero proceed to Step 2 otherwise conclude that the matrix is not invertible.

  2. Step 2: Find the adjoint of the matrix A.

  3. Step 3: Obtain the inverse of A by using the formula .

For the matrix we found the adjoint in the previous example we will find the inverse.

Step 1: To make sure the matrix is invertible we have to check whether the determinant is nonzero.

For the same let us find the value of its determinant. Choosing the first row for the calculation of the value of the determinant, we have the cofactor of its elements to be -13, 26 and-13, calculated in the above example.

Since the determinant is zero the matrix is not invertible.

Calculate the inverse of the matrix .

Solution

Step 1: Calculate the determinant and verify it is nonzero.

Step 2: Find the adjoint of the matrix. To do this we first formulate the cofactor of the matrix.

Now we can obtain the adjoint by taking the transpose of the above matrix which gives us

Step 3: Now we can obtain the inverse of the matrix A

Determinant of Inverse Matrix - Key takeaways

  • Determinant of a matrix: For a square matrix of order 2 - determinant is equal to the subtraction of the product of off-diagonal elements from the product of the main diagonal elements. For a square matrix of order 3 or higher - determinant is equal to the sum of the product of the elements of a row or column with its corresponding cofactor.
  • The necessary and sufficient condition for finding an inverse of a matrix is that its determinant is nonzero
  • Inverse of a matrix: The inverse of a matrix whose determinant is nonzero is given by the formula .

  • Determinants and inverse can be found only for square matrices.

Frequently Asked Questions about Determinant of Inverse Matrix

To find the inverse determinant, we first need to find the inverse of the matrix at hand and then work out its determinant.

You find the determinant of an invertible matrix like you do to any other square matrix. For a square matrix A, det(A) is

.

Yes, generally, the determinant of a square matrix does not equal the determinant of its inverse matrix.

The inverse determinant is the determinant of the inverse matrix of a matrix.

The formula of the inverse matrix of a matrix A is

.

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