Cofactor in Matrix – Detailed Explanation, Inverse of a Matrix, and FAQs

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What is a Cofactor Matrix?

The matrix obtained by removing a row and a column from the matrix is called a cofactor Matrix. Let us understand this in a better way by using an example! 

How Do You Find the Cofactor Matrix?

To find the cofactor Matrix, you need to take each element and remove each row and column. The 4 other elements which are left would come together and constitute the cofactor Matrix

Consider the matrix given below. 

\[\begin{bmatrix} 1 & 2 & 6\\ 4 & 3 & 8\\ 4 & 5 & 6 \end{bmatrix}\]

The cofactor matrices for each element are as given below. 

The cofactor Matrix of each element of the matrix given above are as follows 

Cofactor Matrix with Respect to 1

\[\begin{bmatrix} 3 & 8\\ 5 & 6 \end{bmatrix}\]

Cofactor Matrix with Respect to 2 

\[\begin{bmatrix} 4 & 8\\ 4 & 6 \end{bmatrix}\]

Cofactor Matrix with Respect to 6 

\[\begin{bmatrix} 4 & 8\\ 4 & 6 \end{bmatrix}\]

Cofactor Matrix with Respect to 4

\[\begin{bmatrix} 2 & 6\\ 5 & 6 \end{bmatrix}\]

Cofactor Matrix with Respect to 3

\[\begin{bmatrix} 4 & 6\\ 1 & 6 \end{bmatrix}\]

Cofactor Matrix with Respect to 8

\[\begin{bmatrix} 1 & 2\\ 4 & 5 \end{bmatrix}\]

Cofactor Matrix with Respect to 4 

\[\begin{bmatrix} 2 & 6\\ 3 & 8 \end{bmatrix}\]

Cofactor Matrix with Respect to 5

\[\begin{bmatrix} 1 & 6\\ 4 & 8 \end{bmatrix}\]

Cofactor Matrix with Respect to 6

So from the above example, we can easily notice that each element of a matrix has its own, unique cofactor Matrix. Hence, there are 9 cofactor matrices for a 3×3 matrix.

What is a Minor?

The determinant of a cofactor Matrix is called the minor of a matrix. For instance, consider the matrix given below

\[\begin{bmatrix} 5 & 9\\ 7 & 2 \end{bmatrix}\]

 minor of the matrix is (5×2)-(9×7)= -53

What are Cofactors?

Cofactor definition goes something like this, cofactors are the determinants of the cofactor Matrix along with the sign of the placeholder number with respect to whom the cofactor Matrix is found. Sounds confusing right? Well, let us look at this with an example!.

Consider the 3×3 matrix given below!

\[\begin{bmatrix} 5 & 9 & 6\\ 7 & 2 & 7\\ 4 & 6 & 8 \end{bmatrix}\]

Now, let us find the cofactor matrix of the element 5 from the matrix given above. 

So, the cofactor matrix with respect to element 5 is 

\[\begin{bmatrix} 2 & 7\\ 6 & 8 \end{bmatrix}\]

The determinant of the cofactor matrix is as follows

(8×2)-(7×6) = 26

Now, as we’ve seen above, 26 is just the minor of element 5. However, to find the cofactor you need to go a bit further. You also need to add the sign of the element to the minor. Let us understand this In a better way! 

\[\begin{bmatrix} + & – & +\\ – & + & -\\ + & – & + \end{bmatrix}\]

Above are the signs of each place. While finding the Cofactor, you need to attach the sign of the place at which the element is present. So since 5 is present at the position (1,1) the sign at that position is + and hence a + sign is added to the minor of the element at (1,1). Similarly for the Cofactor of 9 which is present at (1,2) a negative (-) must be attached! 

The cofactor of 5 in the matrix given above is 2. Similarly, the cofactor of the element ‘9’ in the matrix given above is 7. Hence, each element in a matrix is a cofactor to another element in the same matrix!

What is the Inverse of a Matrix? 

The inverse of a matrix is defined as a matrix which when multiplied with the original matrix gives 1. The definition sounded confusing, right? Here’s an easier explanation! 

Suppose that A is a matrix and B is the inverse matrix of A. In this scenario, 

A×B will be equal to 1 since A and B are inverse matrices of each other. The cofactor matrix helps in finding the inverse matrix of the matrix! Therefore, you must remember all about cofactor Matrices while finding an inverse of the matrix. 

Let us implement all that we understood today and try to do a problem! 

Example 1: Find the cofactor of any 4 elements of the matrix given below

\[\begin{bmatrix} 6 & 8 & 9\\ 7 & 5 & 7\\ 2 & 1 & 0 \end{bmatrix}\]

With respect to 6

\[\begin{bmatrix} 5 & 7\\ 1 & 0 \end{bmatrix}\]

The determinant of the matrix= 0-7

Minor is -7 

Since 6 is present at (1,1) the sign is + and hence minor=Cofactor=-7

With respect to 8 

\[\begin{bmatrix} 7 & 7\\ 2 & 0 \end{bmatrix}\]

The determinant of the matrix= 0-14 

Minor is -14

Since 8 is present at a negative placeholder a negative sign is supposed to be added. Hence Cofactor= 14

With respect to 0

\[\begin{bmatrix} 6 & 8\\ 7 & 5 \end{bmatrix}\]

The determinant of the matrix is (6×5)-(8×7) = -26

Minor is -26 

The place is + and hence cofactor=Matrix= -26

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