Cofactor Matrix

The co-factor matrix is formed with the co-factors of the elements of the given matrix. The co-factor 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 co-factor 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 co-factor matrix and the applications of the co-factor matrix.

Co-factor matrix is a matrix having the co-factors as the elements of the matrix. First, let us understand more about the co-factor of an element within the matrix. Co-factor 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 co-factor of the element is denoted as \(C_{ij}\). If the minor of the element is \(M_{ij}\), then the co-factor 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 co-factor, to obtain the co-factor 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\(= \left[\begin{array}{ccc}
M_{11} & M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23} \\
M_{31} & M_{32} & M_{33}
\end{array}\right] \)

\( \begin{align}\text{Co-factor Matrix}&=\left[\begin{array}{ccc}
(-1)^{1 + 1}M_{11} & (-1)^{1 + 2}M_{12} & (-1)^{1 + 3}M_{13} \\
(-1)^{2 + 1}M_{21} & (-1)^{2 + 2}M_{22} & (-1)^{2 + 3}M_{23} \\
(-1)^{3 + 1}M_{31} & (-1)^{3 + 2}M_{32} & (-1)^{3 + 3}M_{33}
\end{array}\right] \\&=\left[\begin{array}{ccc}
+M_{11} & -M_{12} & +M_{13} \\
-M_{21} & +M_{22} & -M_{23} \\
+M_{31} & -M_{32} & +M_{33}
\end{array}\right] \\& = \left[\begin{array}{ccc}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{array}\right] \end{align}\)

The following four simple steps are helpful to find the co-factor 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 co-factor 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 co-factors of the elements of the given matrix, to form the co-factor matrix.

\(A =\begin{bmatrix}a_{11} & a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\)

Co-factor of \(a_{11} = C_{11} =(-1)^{1 + 1}\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right| =+( a_{22}.a_{33} - a_{23}.a_{32})\)

Co-factor of \(a_{23} = C_{23} =(-1)^{2 + 3}\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{31} & a_{32}
\end{array}\right| =-( a_{11}.a_{32} - a_{12}.a_{31})\)

Co-factor 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 co-factor of each element of the matrix A. Further we can form the co-factor matrix of A by writing the co-factor of each element in the matrix array.

Co-factor Matrix of A = \(\begin{bmatrix}C_{11} & C_{12}&C_{13}\\C_{21}&C_{22}&C_{23}\\C_{31}&C_{32}&C_{33}\end{bmatrix}\)

The following are the important applications of the co-factor matrix. The co-factor matrix is helpful to find the adjoint of the matrix and the inverse of the matrix. Also, the co-factors 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 co-factor 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 co-factors. 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 co-factor matrix of the given matrix, and then the transpose of this co-factor 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 co-factor 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 co-factor matrix to obtain the adjoint of the matrix.

Adj A = Transpose of Co-factor 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 Co-factor 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 co-factor matrix.

• Square Matrix
• Types of Matrices
• Matrix Formula
• Transpose of a Matrix
• Adjoint of a Matrix
• Inverse of a Matrix
• Symmetric Matrix
• Skew Symmetric Matrix

What is Co-factor Matrix?

Co-factor matrix is a matrix having the co-factors as the elements of the matrix. Co-factor 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 co-factor of the element is denoted as \(C_{ij}\). If the minor of the element is \(M_{ij}\), then the co-factor of element would be:

\(C_{ij} = (-1)^{i+j}) M_{ij}\)

How to Find the Co-factor Matrix?

The following four simple steps are helpful to find the co-factor 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 co-factor 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 co-factors of the elements of the given matrix, to form the co-factor matrix.

How to Find the Co-factor Matrix of a 2 × 2 Matrix?

The co-factor 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 co-factor matrix of A = \(\begin{bmatrix}d & -c\\-b&a\end{bmatrix}\)

How to Convert Co-factor Matrix to the Adjoint of a Matrix?

The transpose of the co-factor matrix gives the adjoint matrix. The formula for transpose of a matrix is used to find the transpose of the co-factor matrix.

Adj A = Transpose of Co-factor 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 Co-factor Matrix?

The co-factor matrix is helpful to find the adjoint of the matrix and the inverse of the matrix. Also the co-factors of the elements of the matrix are useful in the calculation of determinant of the matrix.