What is Cofactor in Matrix Formula Steps and Solved Examples
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
FAQs on Cofactor in Matrix Explained with Definition and Formula
1. What is a cofactor in a matrix?
A cofactor of an element in a matrix is the signed minor of that element used in determinant and inverse calculations. For an element aij in a square matrix, its cofactor is defined as Cij = (−1)i+j Mij, where Mij is the minor obtained by deleting the i-th row and j-th column. Cofactors are mainly used in finding the determinant and the adjugate (adjoint) matrix.
2. How do you find the cofactor of an element in a matrix?
To find the cofactor of an element, delete its row and column, find the minor, and apply the sign factor (−1)i+j. Follow these steps:
- Choose element aij.
- Delete the i-th row and j-th column to get the minor Mij.
- Compute the determinant of the remaining matrix.
- Multiply by (−1)i+j to get Cij.
3. What is the formula for cofactor in a matrix?
The formula for the cofactor of element aij is Cij = (−1)i+j Mij. Here:
- Mij is the minor (determinant after deleting row i and column j).
- (−1)i+j gives the correct sign.
4. What is the difference between a minor and a cofactor?
The minor is the determinant obtained after deleting a row and column, while the cofactor is the signed minor. Specifically:
- Minor (Mij): Determinant after removing row i and column j.
- Cofactor (Cij): (−1)i+j × Mij.
5. How do you find the determinant using cofactor expansion?
The determinant of a matrix can be found by multiplying elements of a row or column by their cofactors and adding the results. For a 3×3 matrix, choose any row i and compute:
- det(A) = ai1Ci1 + ai2Ci2 + ai3Ci3.
6. What is the cofactor matrix?
The cofactor matrix is the matrix formed by replacing each element of a square matrix with its corresponding cofactor. If A = [aij], then the cofactor matrix is [Cij]. It is used to find the adjugate (adjoint) matrix, which is the transpose of the cofactor matrix.
7. How is the adjoint (adjugate) of a matrix related to cofactors?
The adjoint (adjugate) of a matrix is the transpose of its cofactor matrix. Mathematically, adj(A) = [Cij]T. Steps:
- Find all cofactors Cij.
- Form the cofactor matrix.
- Take its transpose.
8. How do you find the inverse of a matrix using cofactors?
The inverse of a matrix using cofactors is given by A−1 = (1/det(A)) adj(A), provided det(A) ≠ 0. Steps:
- Find det(A).
- Compute all cofactors to form the cofactor matrix.
- Take the transpose to get adj(A).
- Multiply adj(A) by 1/det(A).
9. Can you give an example of finding a cofactor in a 2×2 matrix?
Yes, in a 2×2 matrix, the cofactor is found using the same formula Cij = (−1)i+jMij. Example: Let A = [[1, 2], [3, 4]].
- For element 1 (position 1,1), delete row 1 and column 1 → remaining element is 4.
- Minor M11 = 4.
- C11 = (−1)1+1 × 4 = 4.
10. Why is the sign (−1)i+j used in the cofactor formula?
The sign factor (−1)i+j ensures the correct alternating sign pattern in determinant expansion. It creates a checkerboard pattern of signs:
- + − +
- − + −
- + − +
