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
1. What is the definition of a cofactor for an element in a matrix?
A cofactor is a signed numerical value associated with a specific element in a square matrix. It is calculated by finding the minor of the element and then multiplying it by a sign (+1 or -1) determined by the element's position (row and column) within the matrix.
2. What is the formula used to find the cofactor of an element?
The formula to calculate the cofactor of an element aᵢⱼ (located in the i-th row and j-th column) is:
Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ
Where:
- Cᵢⱼ is the cofactor of the element.
- (i+j) represents the sum of the row and column positions.
- Mᵢⱼ is the minor of the element, which is the determinant of the sub-matrix formed by removing the i-th row and j-th column.
3. What is the main difference between a minor and a cofactor in a matrix?
The key difference lies in the sign. A minor is simply the determinant of the sub-matrix obtained by deleting the row and column of a specific element. A cofactor is the very same minor but with an applied sign (+ or -) based on its position. In short, the cofactor incorporates positional sign information, whereas the minor does not.
4. How do you calculate the cofactor of an element in a 3x3 matrix?
To find the cofactor of any element in a 3x3 matrix, follow these steps:
- First, select the element. For example, the element in the first row and first column, a₁₁
- Temporarily remove the entire row and column that the element belongs to. This will leave you with a 2x2 sub-matrix.
- Calculate the determinant of this 2x2 sub-matrix. This value is the minor.
- Determine the sign using the formula (-1)ⁱ⁺ʲ. For a₁₁, the sign is (-1)¹⁺¹ = +1.
- Multiply the minor by this sign to get the final cofactor.
5. What is the primary application of cofactors in matrix algebra as per the CBSE Class 12 syllabus for 2025-26?
The most important application of cofactors is to calculate the adjoint and the inverse of a square matrix. These operations are fundamental for solving systems of linear equations using the matrix method, a key topic in the CBSE Class 12 Maths syllabus.
6. How do cofactors help in finding the adjoint and inverse of a matrix?
The relationship is direct and sequential:
1. First, you calculate the cofactor for every single element in the original matrix.
2. You then arrange these cofactors into a new matrix, called the Matrix of Cofactors.
3. The adjoint of the original matrix is the transpose of this Matrix of Cofactors.
4. Finally, the inverse is calculated by dividing the adjoint matrix by the determinant of the original matrix.
7. Is it possible to find cofactors for a non-square (rectangular) matrix?
No, the concepts of minors and cofactors are defined exclusively for square matrices (e.g., 2x2, 3x3, etc.). This is because the process relies on calculating a determinant, an operation that is only applicable to square matrices.
8. What is the logic behind the alternating sign pattern used for cofactors?
The alternating sign pattern is not arbitrary; it comes from the formula (-1)ⁱ⁺ʲ. The sign depends on whether the sum of the row number (i) and column number (j) is even or odd.
- If (i+j) is even, (-1) raised to an even power is +1.
- If (i+j) is odd, (-1) raised to an odd power is -1.
