Minor of Matrix

The minor of matrix is for each element of matrix and is equal to the part of the matrix remaining after excluding the row and the column containing that particular element. The new matrix formed with the minors of each element of the given matrix is called the minor of matrix.

The minor of matrix is prominently used in finding its determinant, adjoint, and inverse of a matrix. Let us learn more about the minor of matrix in the below content.

Minor of matrix for a particular element in the matrix is defined as the matrix obtained after deleting the row and column of the matrix in which that particular element lies. Here the minor of the element \(a_{ij}\) is denoted as \(M_{ij}\). For example, for the given matrix A, the minor of \(a_{12}\) is the part of the matrix after excluding the first row and the second column of the 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 take the minors of the matrix and form a minor matrix M of the given matrix A as:

\(M = \left[\begin{array}{ccc}
M_{11} & M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23} \\
M_{31} & M_{32} & M_{33}
\end{array}\right] \)

There are three simple steps to find the minor of the matrix.

• First identity and exclude the row and the column which contains the particular element within the matrix.
• As a second step, form a new smaller matrix with the remaining elements, to represent the minor of the particular element of the matrix.
• Finally, find the determinant of the minor of each element of the matrix, and form a new matrix containing the minor values of the respective elements.

This creates the minor of matrix.

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

Minor of \(a_{11} = M_{11} =\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right| = a_{22}.a_{33} - a_{23}.a_{32}\)

Minor of \(a_{23} = M_{23} =\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{31} & a_{32}
\end{array}\right| = a_{11}.a_{32} - a_{12}.a_{31}\)

Minor of \(a_{32} = M_{23} =\left|\begin{array}{ll}
a_{11} & a_{13} \\
a_{21} & a_{23}
\end{array}\right| = a_{11}.a_{23} - a_{13}.a_{21}\)

Similary we can find the minor of each element of the matrix A. Further we can form the minor of matrix A by writing the minor of each element in the matrix array.

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

The minor of the matrix is useful to find the cofactors of the elements of the matrix, which is helpful to find the adjoint of the matrix, and the inverse of the matrix. Also minor of the matrix is used in the calculation of determinant of the matrix. Let us now try to understand the following important applications of the minor of the matrix.

Cofactor Matrix

Cofactor of an element in matrix A is obtained when the minor \(M_{ij}\) of the element is multiplied with (-1)^i+j. The cofactor of an element is denoted as \(C_{ij}\). If the minor of a matrix is \(M_{ij}\), then the cofactor of the element would be:

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

The matrix formed with the cofactors of the elements of the matrix and is called the cofactor matrix.

Cofactor Matrix = \(\left[\begin{array}{ccc}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{array}\right] \)

This cofactor matrix is being referred to the below matrix A.

Determinant of a matrix

The determinant of a matrix is a summary value and is calculated using the cofactors of 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. Let say consider the matrix A.

\(A = \left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right] \)

Determinant of a matrix A is denoted as |A|. Here we calculate the determinant with respect to the elements of the first row of the matrix. Then the 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 a matrix of this cofactor matrix to obtain the adjoint of a matrix. Let us consider the following matrix A.

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

Cofactor matrix of \(A = \begin{bmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}\).

Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{bmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}\) =\(\begin{bmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{bmatrix}\)

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

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

A^-1 = \(\dfrac{1}{|A|}\). \(\begin{bmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{bmatrix}\)

Related Topics

The following related topics are helpful for a better understanding of this concept of minor of matrix.

• Square Matrix
• Types of Matrices
• Matrix Formula
• Transpose of a Matrix

What is Minor of Matrix?

The minor of matrix is for each element of matrix and is equal to the part of the matrix remaining after excluding the row and the column containing that particular element. The minor of matrix is defined only for a square matrix. The minor of the element 'a' in the matrix A = \(\begin{bmatrix}a & b\\c&d\end{bmatrix}\) is d.

How to Find Minors of Matrix?

There are two simple steps to find the minor of the matrix. First identity and exclude the row and the column which contains the particular element within the matrix. Next, form a new smaller matrix with the remaining elements, to represent the minor of the particular element of the matrix.

The minor of the element 'e' in the matrix A = \(\begin{bmatrix}a&b & c\\d&e&f\\g&h&i\end{bmatrix}\) is M = \(\begin{bmatrix}a & c\\g&i\end{bmatrix}\).

How to Find the Minors of a 2 × 2 Matrix?

For a matrix of order 2 × 2 of the form A = \(\begin{pmatrix}a & b\\c&d\end{pmatrix}\), the minor of matrix A = \(\begin{pmatrix}d & c\\b&a\end{pmatrix}\). The minor of a particular element within the matrix is equal to the remaining element after excluding the row and column containing that particular element.

What Is the Difference Between Minors of Matrix and Cofactor of Matrix?

The minor of an element \(a_{ij}\) is denoted as \(M_{ij}\). The cofactor of a matrix is obtained from the minor of the matrix and is equal to the product of (-1)^{i + j} and the minor of the matrix. Furher, if the minor of a matrix is \(M_{ij}\), then the cofactor of the matrix would be \(C_{ij} = (-1)^{i+j}) M_{ij}\).

What Are the Uses of Minors of Matrix?

The minor of matrix is useful to find the cofactors of the elements of the matrix. The minors of the matrix is used to find the determinant value of the matrix. Further, these minors and cofactors of the matrix can be used to find the determinantof the matrix, adjoint of the matrix, and the inverse of the matrix.