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Cofactor Matrix
The cofactor matrix is formed with the cofactors of the elements of the given matrix. The cofactor of an element of the matrix is equal to the product of the minor of the element and 1 to the power of the positional value of the element.
The cofactor matrix is useful to find the adjoint of the matrix and the inverse of the given matrix. Here we shall learn how to find the cofactor matrix and the applications of the cofactor matrix.
1.  What Is Cofactor Matrix? 
2.  How to Find the Cofactor Matrix? 
3.  Applications of Cofactor Matrix 
4.  Examples on Cofactor Matrix 
5.  Practice Questions 
6.  FAQs on Cofactor Matrix 
What Is Cofactor Matrix?
Cofactor matrix is a matrix having the cofactors as the elements of the matrix. First, let us understand more about the cofactor of an element within the matrix. Cofactor of an element within the matrix is obtained when the minor \(M_{ij}\) of the element is multiplied with (1)^{i+j}. Here i and j are the positional values of the element and refers to the row and the column to which the given element belongs. The cofactor of the element is denoted as \(C_{ij}\). If the minor of the element is \(M_{ij}\), then the cofactor of element would be:
\(C_{ij} = (1)^{i+j}) M_{ij}\)
Here first we need to find the minor of the element of the matrix and then the cofactor, to obtain the cofactor matrix
\(A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right] \)
The minor of the element \(a_{12}\) is as follows.
\(M_{12} = \left[\begin{array}{ccc} a_{21} & a_{23} \\
a_{31} & a_{33}
\end{array}\right] \)
Similarly, we can find all the minors of the matrix and we can form the minor matrix M of the given matrix A as:
Minor Matrix\(= \left[\begin{array}{ccc}
M_{11} & M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23} \\
M_{31} & M_{32} & M_{33}
\end{array}\right] \)
\( \begin{align}\text{Cofactor Matrix}&=\left[\begin{array}{ccc}
(1)^{1 + 1}M_{11} & (1)^{1 + 2}M_{12} & (1)^{1 + 3}M_{13} \\
(1)^{2 + 1}M_{21} & (1)^{2 + 2}M_{22} & (1)^{2 + 3}M_{23} \\
(1)^{3 + 1}M_{31} & (1)^{3 + 2}M_{32} & (1)^{3 + 3}M_{33}
\end{array}\right] \\&=\left[\begin{array}{ccc}
+M_{11} & M_{12} & +M_{13} \\
M_{21} & +M_{22} & M_{23} \\
+M_{31} & M_{32} & +M_{33}
\end{array}\right] \\& = \left[\begin{array}{ccc}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{array}\right] \end{align}\)
How to Find the Cofactor Matrix?
The following four simple steps are helpful to find the cofactor matrix of the given matrix.
 First, find the minor of each element of the matrix by excluding the row and column of that particular element, and then taking the remaining part of the matrix.
 Secondly, find the minor element value by taking the determinant of the remaining part of the matrix.
 .The third step involves finding the cofactor of the element by multiplying the minor of the element with 1 to the power of position values of the element.
 The fourth steps involves forming a new matrix with the cofactors of the elements of the given matrix, to form the cofactor matrix.
\(A =\begin{bmatrix}a_{11} & a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\)
Cofactor of \(a_{11} = C_{11} =(1)^{1 + 1}\left\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right =+( a_{22}.a_{33}  a_{23}.a_{32})\)
Cofactor of \(a_{23} = C_{23} =(1)^{2 + 3}\left\begin{array}{ll}
a_{11} & a_{12} \\
a_{31} & a_{32}
\end{array}\right =( a_{11}.a_{32}  a_{12}.a_{31})\)
Cofactor of \(a_{32} = C_{23} =(1)^{2 + 3}\left\begin{array}{ll}
a_{11} & a_{13} \\
a_{21} & a_{23}
\end{array}\right = (a_{11}.a_{23}  a_{13}.a_{21})\)
Similarly we can find the cofactor of each element of the matrix A. Further we can form the cofactor matrix of A by writing the cofactor of each element in the matrix array.
Cofactor Matrix of A = \(\begin{bmatrix}C_{11} & C_{12}&C_{13}\\C_{21}&C_{22}&C_{23}\\C_{31}&C_{32}&C_{33}\end{bmatrix}\)
Applications of Cofactor Matrix
The following are the important applications of the cofactor matrix. The cofactor matrix is helpful to find the adjoint of the matrix and the inverse of the matrix. Also, the cofactors of the elements of the matrix are useful in the calculation of determinant of the matrix. Let us now try to understand in detail, each of the applications of the cofactor matrix.
Determinant of a matrix
The determinant of a matrix is a summary value and is calculated using the elements of the matrix. Determinant of a matrix is equal to the summation of the product of the elements of a particular row or column with their respective cofactors. The determinant of a matrix is defined only for square matrices. Determinant of a matrix A is denoted as A.
\(A = \left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right] \)
Then determinant formula of matrix A is as follows.
A = \(a_{11}(1)^{1 + 1} \left\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right + a_{12}(1)^{1 + 2} \left\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right + a_{13}(1)^{1 + 3} \left\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{matrix}\right\)
Adjoint of the matrix
The adjoint of a 3 x 3 matrix can be obtained by following two simple steps. First we need to find the cofactor matrix of the given matrix, and then the transpose of this cofactor matrix is taken to obtain the adjoint of a matrix. For a matrix of the form A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the cofactor matrix A = \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\). Further we need to take the transpose of this cofactor matrix to obtain the adjoint of the matrix.
Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
Inverse of a Matrix
The inverse of a matrix can be computed by dividing the adjoint of a matrix by the determinant of the matrix. For a matrix A, its inverse A^{1} = \(\dfrac{1}{A}\).Adj A.
A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\)
A = \(a_{11}(1)^{1 + 1} \left\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right + a_{12}(1)^{1 + 2} \left\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right + a_{13}(1)^{1 + 3} \left\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{matrix}\right\)
Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
A^{1} = \(\dfrac{1}{A}\). \(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
Related Topics
The following related topics are helpful for a better understanding of cofactor matrix.
Examples on Cofactor Matrix

Example 1: Find the cofactor matrix of the matrix \(\begin{bmatrix}4 & 7\\11&9\end{bmatrix}\).
Solution:
The given matrix \(\begin{bmatrix}4 & 7\\11&9\end{bmatrix}\) represents a 2 × 2 matrix.
For a matrix A = \(\begin{bmatrix}a & b\\c&d\end{bmatrix}\), the cofactor matrix of A = \(\begin{bmatrix}d & c\\b&a\end{bmatrix}\)
Hence the cofactor matrix of the given matrix is = \(\begin{bmatrix}9 & 11\\7&4\end{bmatrix}\)
Answer: Therefore the cofactor matrix is \(\begin{bmatrix}9 & 11\\7&4\end{bmatrix}\).

Example 2: Find the cofactor matrix and the adjoint matrix for the given matrix \(\begin{bmatrix}5 & 9&2\\1 &8& 5\\3&6&4\end{bmatrix}\).
Solution:
The given matrix is \(\begin{bmatrix}5 & 9&2\\1 &8& 5\\3&6&4\end{bmatrix}\).
Let us now first find the cofactors of each of the elements of the above matrix.
Cofactor of \(5 =(1)^{1 + 1}\left\begin{array}{ll}
8 & 5 \\
6 & 4
\end{array}\right =+( 8(4)  6(5)) = 32  30 = 2\)Cofactor of \(9 =(1)^{1 + 2}\left\begin{array}{ll}
1 & 5 \\
3 & 4
\end{array}\right =( 1(4)  3(5)) = (4  15) = 11\)Cofactor of \(2 =(1)^{1 + 3}\left\begin{array}{ll}
1 & 8 \\
3 & 6
\end{array}\right =+( 1(6)  3(8)) = 6  24 = 18\)Cofactor of \(1=(1)^{2 + 1}\left\begin{array}{ll}
9 & 2 \\
6 & 4
\end{array}\right =( 9(4)  2(6)) = (36  12) = 24\)Cofactor of \(8 =(1)^{2 + 2}\left\begin{array}{ll}
5 & 2 \\
3 & 4
\end{array}\right =+( 5(4)  3(2)) = 20  6 = 14\)Cofactor of \(5 =(1)^{2 + 3}\left\begin{array}{ll}
5 & 9 \\
3 & 6
\end{array}\right =( 5(6)  9(3)) = (30  27) = 3\)Cofactor of \(3 =(1)^{3 + 1}\left\begin{array}{ll}
9 & 2 \\
8 & 5
\end{array}\right =+( 9(5)  2(8)) = 45  16 = 29\)Cofactor of \(6 =(1)^{3 + 2}\left\begin{array}{ll}
5 & 2 \\
1 & 5
\end{array}\right =( 5(5)  1(2)) = (25  2) = 23\)Cofactor of \(4 =(1)^{3 + 3}\left\begin{array}{ll}
5 & 9 \\
1 & 8
\end{array}\right =+( 5(8)  1(9)) = 40  9 = 31\)Cofactor Matrix = \(\begin{bmatrix}2 & 11&`18\\24 &14& 3\\29&23&31\end{bmatrix}\)
Adjoint Matrix = \(\begin{bmatrix}2 & 24&29\\11 &14& 23\\18&3&31\end{bmatrix}\)
Answer: Therefore the cofactor matrix is \(\begin{bmatrix}2 & 11&`18\\24 &14& 3\\29&23&31\end{bmatrix}\), and adjoint Matrix is \(\begin{bmatrix}2 & 24&29\\11 &14& 23\\18&3&31\end{bmatrix}\).
FAQs on Cofactor Matrix
What is Cofactor Matrix?
Cofactor matrix is a matrix having the cofactors as the elements of the matrix. Cofactor of an element within the matrix is obtained when the minor \(M_{ij}\) of the element is multiplied with (1)^{i+j}. Here i and j are the positional values of the element and refers to the row and the column to which the given element belongs. The cofactor of the element is denoted as \(C_{ij}\). If the minor of the element is \(M_{ij}\), then the cofactor of element would be:
\(C_{ij} = (1)^{i+j}) M_{ij}\)
How to Find the Cofactor Matrix?
The following four simple steps are helpful to find the cofactor matrix of the given matrix.
 First, find the minor of each element of the matrix by excluding the row and column of that particular element, and then taking the remaining part of the matrix.
 Secondly, find the minor element value by taking the determinant of the remaining part of the matrix.
 .The third step involves finding the cofactor of the element by multiplying the minor of the element with 1 to the exponent of position values of the element.
 The fourth steps involves forming a new matrix with the cofactors of the elements of the given matrix, to form the cofactor matrix.
How to Find the Cofactor Matrix of a 2 × 2 Matrix?
The cofactor matrix of a 2 x 2 matrix can be defined by using a formula. For a matrix A = \(\begin{bmatrix}a & b\\c&d\end{bmatrix}\), the cofactor matrix of A = \(\begin{bmatrix}d & c\\b&a\end{bmatrix}\)
How to Convert Cofactor Matrix to the Adjoint of a Matrix?
The transpose of the cofactor matrix gives the adjoint matrix. The formula for transpose of a matrix is used to find the transpose of the cofactor matrix.
Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)
What Are the Applications of Cofactor Matrix?
The cofactor matrix is helpful to find the adjoint of the matrix and the inverse of the matrix. Also the cofactors of the elements of the matrix are useful in the calculation of determinant of the matrix.
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