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Minors And Cofactors
Minors and cofactors can be computed for each of the elements of a matrix. Minor of an element is equal to the determinant of the remaining elements of the matrix, after excluding the row and column containing the particular element. The cofactor of an element can be calculated from the minor of the element. The cofactor of an element is equal to the product of minor of the element, and 1 to the power of position values of row and column of the element.
Cofactor of an Element = (1)^{i + j }× Minor of an Element
Here i and j are the positional values of the row and column of the element. Let us learn more about the minors and cofactors, and their applications, with the help of examples, FAQs.
What Are Minors And Cofactors?
Minors and cofactors are defined for each element of the matrix. The minor of an element of the matrix is equal to the determinant of the remaining elements of the matrix, obtained after deleting the row and column containing the particular element in the matrix. The cofactor of an element of the matrix is obtained by multiplying the minor of the element with 1 to the exponent(power) of the sum of i^{th }row and j^{th }column containing the element.
A = \(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i \end{bmatrix}\)
Minor of a = \(\begin{bmatrix}e&f\\h&i \end{bmatrix}\)
The minor of the first element of the first row of the above matrix A has been obtained after ignoring the first row and first column of the above matrix and forming a new matrix. Further, the cofactor of the element a is obtained by multiplying the minor with (1) to the power of the position value row and column of element a.
Cofactor of a = (1)^{i + j} × Minor of a
Cofactor of a = (1)^{1 + 1}× \(\begin{bmatrix}e&f\\h&i \end{bmatrix}\)
The minor and cofactor of a matrix are simple numeric values, which are obtained after taking the determinant of the remaining elements of the given matrix.
Comparison Of Minors And Cofactors
The cofactor is obtained from the minor of the element of a matrix. The cofactor is equal to the product of the minor of the element of the matrix, and 1 to the power of the sum of the positional value of the row and column containing the element. The numeric value of the minor or the cofactor of an element of the matrix is equal but only differs by a sign, which is dependent on (1)^{i + j}.
Cofactor of an Element = (1)^{i + j}× Minor of an Element
Here i and j are the positional values and refer to the row and the column to which the element belongs. The minor and cofactor of an element of the matrix may have the same or different sign. The minor is used only to find the determinant of the matrix, and the cofactor is used to find the adjoint and the inverse of a matrix.
Applications Of Minors And Cofactors
The minors and cofactors are useful to find the adjoint and inverse of a matrix. The adjoint of a matrix is equal to the transpose of the cofactors of the elements of the given matrix. The inverse of a matrix is equal to the adjoint of the matrix divided by the determinant of the matrix. Further, the determinant of the matrix is equal to the summation of the product of the elements of any row or column of the matrix with their respective cofactors.
Let us learn about the application of minors and cofactors to find the determinant, adjoint, and inverse of a 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
Examples on Minors And Cofactors

Example 1: Finding the minor and cofactor of 7 in the matrix \(\begin{bmatrix}3&7&4\\2&6&3\\3&5&1\end{bmatrix}\).
Solution:
The given matrix is A = \(\begin{bmatrix}3&7&4\\2&6&3\\3&5&1\end{bmatrix}\)
We need to find the cofactor of 7
Cofactor of 7 = \((1)^{1 + 2}\begin{bmatrix}2&3\\3&1\end{bmatrix}\) = \((1)^3[2(1)  3(3)]\) = (1)[2 + 9] = (11) = 11
Therefore the cofactor of 7 is 11.

Example 2: Find the cofactor matrix for the matrix \(\begin{bmatrix}3&0&4\\2&1&3\\3&5&1\end{bmatrix}\)
Solution:
The given matrix is A = \(\begin{bmatrix}3&0&4\\2&1&3\\3&5&1\end{bmatrix}\).
Cofactor of 3 = \((1)^{1 + 1} \times \begin{bmatrix}1&3\\5&1\end{bmatrix}\) = \((1)^2 [(1)×1  3×5]\) = 1[1  15] = 16
Cofactor of 0 = \((1)^{1 + 2} \times \begin{bmatrix}2&3\\3&1\end{bmatrix}\) = \((1)^3 [2×1  3×(3)]\) = 1[2 +9] = 11
Cofactor of 4 = \((1)^{1 + 3} \times \begin{bmatrix}2&1\\3&5\end{bmatrix}\) = \((1)^4 [2×5  (1)×(3)]\) = 1[10  3] = 7
Cofactor of 2 = \((1)^{2 + 1} \times \begin{bmatrix}0&4\\5&1\end{bmatrix}\) = \((1)^3 [0×1  4×5]\) = 1[0  20] = +20
Cofactor of 1 = \((1)^{2 + 2} \times \begin{bmatrix}3&4\\3&1\end{bmatrix}\) = \((1)^4 [3×1  (3)×4]\) = 1[3 + 12] = +15
Cofactor of 3 = \((1)^{2 + 3} \times \begin{bmatrix}3&0\\3&5\end{bmatrix}\) = \((1)^5 [3×5  0×(3)]\) = 1[15  0] = 15
Cofactor of 3 = \((1)^{3 + 1} \times \begin{bmatrix}0&4\\1&3\end{bmatrix}\) = \((1)^4 [0×3  4×(1)]\) = 1[0 +4] = 4
Cofactor of 5 = \((1)^{3 + 2} \times \begin{bmatrix}3&4\\2&3\end{bmatrix}\) = \((1)^5 [3×3  4×2]\) = 1[9  8] = 1
Cofactor of 1 = \((1)^{3 + 3} \times \begin{bmatrix}3&0\\2&1\end{bmatrix}\) = \((1)^6 [3×(1)  0×2]\) = 1[3  0] = 3
Therefore, the Cofactor Matrix of given matrix A = \( \begin{bmatrix}16&11&7\\+20&+15&15\\4&1&3\end{bmatrix}\)
FAQs on Minors And Cofactors
What Are Minors And Cofactors?
Minors and cofactors are the representative values of each of the elements of a matrix. The minor is obtained by taking the determinant of the elements remaining after excluding the row and the column of the particular element. The cofactor is obtained from the minor and is equal to the product of the minor of the element with 1 to the exponent of sum of the position values of the row and the column.
How To Find Minors And Cofactors?
The minor and cofactors can be computed for each of the elements. The minor of an element is equal to the determinant of the matrix remaining after excluding the row and column containing the element. The cofactor of an element is equal to the product of the minor of the element, and 1 to the power of the row and column of the element.
Cofactor of an Element = (1)^{i + j}× Minor of an Element
What Is The Formula For Minors And Cofactors?
The formula for minor and cofactor of an element is Cofactor of an Element = (1)^{i + j}× Minor of an Element. Here i and j are the positional value of the row and column of the element.
Are Minors And Cofactors The Same?
The minors and cofactors are different, and the cofactor can be obtained from the minor of the element. The cofactor of the element is equal to the product of the minor of the element, and 1 to the power of row and column of the element.
What Is The Order Of Minors And Cofactor?
The minors and cofactors do not have any order. The minors and cofactors are simple numeric values, which have been computed from the determinant.
What Is The Difference Between Minors And Cofactors?
The minors and cofactors differ by the sign of the value. The cofactor can be computed from the minor of the matrix. The cofactor is equal to the product of the minor of the element, and 1 to the sum of the row and column of the element.
What Are The Uses Of Minors And Cofactors?
The minors and cofactors are useful to the determinant of the matrix, adjoin of the matrix, and the inverse of the matrix.
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