A minor of a matrix is the determinant of a smaller square matrix that you get when you remove one or more rows and columns from a larger matrix. It helps in doing bigger calculations, like finding the determinant or inverse of a matrix.

A matrix is just a way to organize numbers, symbols, or expressions in rows and columns. It looks like a box or table and is very useful in many areas of math and science. Each item in a matrix is called an element.

Matrices are used in many ways:

• In geometry, to represent shapes, movements, and changes in position.
• In computer graphics, to show 3D objects on a 2D screen.
• In mathematics, to solve sets of equations and to check if three points lie on the same line (collinear points).

The order of a matrix tells you how big it is — it's written as the number of rows × number of columns.
For example, a 2 × 2 matrix has 2 rows and 2 columns, and so it contains 4 elements in total.

Minors play a key role in understanding more complex matrix operations.

What is Minor of a Matrix?

The minor of an element in a matrix is the value you get by removing the row and column that the element is in, and then finding the determinant of the smaller matrix that remains.

In a matrix, every element has its own minor. If the matrix is called A, and you're finding the minor of the element in the i-th row and j-th column (written as aij), you do this:

1. Remove the i-th row and j-th column from the matrix A.
2. Use the remaining numbers to form a new smaller square matrix.
3. Find the determinant of that smaller matrix — that result is the minor of aij.

When you calculate the minor for each element of a matrix, and use those values to form a new matrix, this new matrix is called the minor matrix, and it’s usually written as M.

Minor of a Matrix Formula

The formula to find the minor of a matrix is given below:

Let us consider a 3×3 matrix, \( A=\begin{bmatrix}a&\ b&\ c\\d&\ e&\ f\\g&\ h&\ i\end{bmatrix} \)

So we pick one element at a time and find the minor of that element by eliminating the row and columns to which that element belongs.

Here,

Minor of \( a \),

\( M_a=\begin{bmatrix}\cancel{a}&\ \cancel{b}&\ \cancel{c}\\\cancel{d}&\ e&\ f\\\cancel{g}&\ h&\ i\end{bmatrix}=\begin{bmatrix}e&f\\h&i\end{bmatrix}=ei-fh \)

Similarly,

\( M_b=\begin{bmatrix}\cancel{a}&\ \cancel{b}&\ \cancel{c}\\d&\ \cancel{e}&\ f\\g&\ \cancel{h}&\ i\\\end{bmatrix}=\begin{bmatrix}d&f\\g&i\end{bmatrix}=di-fg \)

\( M_c=\begin{bmatrix}\cancel{a}&\ \cancel{b}&\ \cancel{c}\\d&\ e&\ \cancel{f}\\g&\ h&\ \cancel{i}\end{bmatrix}=\begin{bmatrix}d&e\\g&h\end{bmatrix}=dh-eg \)

\( M_d=\begin{bmatrix}\cancel{a}&\ b&\ c\\\cancel{d}&\ \cancel{e}&\ \cancel{f}\\\cancel{g}&\ h&\ i\end{bmatrix}=\begin{bmatrix}b&c\\h&i\end{bmatrix}=bi-ch \)

\( M_e=\begin{bmatrix}a&\ \cancel{b}&\ c\\\cancel{d}&\ \cancel{e}&\ \cancel{f}\\g&\ \cancel{h}&\ i\end{bmatrix}=\begin{bmatrix}a&c\\g&i\end{bmatrix}=ai-cg \)

\( M_f=\begin{bmatrix}a&\ b&\ \cancel{c}\\\cancel{d}&\ \cancel{e}&\ \cancel{f}\\g&\ h&\ \cancel{i}\end{bmatrix}=\begin{bmatrix}a&b\\g&h\end{bmatrix}=ah-bg \)

\( M_g=\begin{bmatrix}a&\ b&\ \cancel{c}\\d&\ e&\ \cancel{f}\\\cancel{g}&\ \cancel{h}&\ \cancel{i}\end{bmatrix}=\begin{bmatrix}a&b\\d&e\end{bmatrix}=ae-bd \)

\( M_h=\begin{bmatrix}a&\ \cancel{b}&\ c\\d&\ \cancel{e}&\ f\\\cancel{g}&\ \cancel{h}&\ \cancel{i}\end{bmatrix}=\begin{bmatrix}a&c\\d&f\end{bmatrix}=af-cd \)

\( M_i=\begin{bmatrix}a&\ b&\ \cancel{c}\\d&\ e&\ \cancel{f}\\\cancel{g}&\ \cancel{h}&\ \cancel{i}\end{bmatrix}=\begin{bmatrix}a&b\\d&e\end{bmatrix}=ae-bd \)

Now a 2×2 matrix case is trivial as the determinant is simply the single element left after removing the row and column and the formula is much simpler which is as follows:

If we are given a matrix \( A=\begin{bmatrix}a_{11}&\ a_{12}\\a_{21}&\ a_{22}\end{bmatrix} \), then the minor will be,

\( M_{11}=a_{22} \)

\( M_{12}=a_{21} \)

\( M_{21}=a_{12} \)

\( M_{22}=a_{11} \)

where \( M_{ij} \) denotes the minor of \( a_{ij} \)

How to Find Minor of a Matrix

The following are the steps to find the minor of a matrix.

• First, eliminate the row and column of the respective element in which it is contained.
• Form a new matrix excluding the deleted elements.
• Finally, find the determinant of the newly formed matrix. This will give us the minor of that particular element of the matrix.
• Repeat this procedure to find the minor of each of the elements of the matrix.
• Form a matrix with minor elements.

Thus we will get the minor of the matrix.

For e.g. we have a matrix, \( A=\begin{bmatrix}a_{11}&\ a_{12}&\ a_{13}\\a_{21}&\ a_{22}&\ a_{23}\\a_{31}&\ a_{32}&\ a_{33}\end{bmatrix} \), then the minors for each of the elements of A is,

\(M_{11}=\left|\begin{bmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{bmatrix}\right|=\left(a_{22}\cdot a_{33}\right)-\left(a_{23}\cdot a_{32}\right) \)

\( M_{12}=\left|\begin{bmatrix}a_{21}&\ a_{23}\\a_{31}&\ a_{33}\end{bmatrix}\right|=\left(a_{21}\cdot a_{33}\right)-\left(a_{23}\cdot a_{31}\right)\)

\( M_{13}=\left|\begin{bmatrix}a_{21}&\ a_{22}\\a_{31}&\ a_{32}\end{bmatrix}\right|=\left(a_{21}\cdot a_{32}\right)-\left(a_{22}\cdot a_{31}\right) \)

\( M_{21}=\left|\begin{bmatrix}a_{12}&\ a_{13}\\a_{32}&\ a_{33}\end{bmatrix}\right|=\left(a_{12}\cdot a_{33}\right)-\left(a_{13}\cdot a_{32}\right) \)

\( M_{22}=\left|\begin{bmatrix}a_{11}&\ a_{13}\\a_{31}&\ a_{33}\end{bmatrix}\right|=\left(a_{11}\cdot a_{33}\right)-\left(a_{13}\cdot a_{31}\right)\)

\( M_{23}=\left|\begin{bmatrix}a_{11}&\ a_{12}\\a_{31}&\ a_{32}\end{bmatrix}\right|=\left(a_{11}\cdot a_{32}\right)-\left(a_{12}\cdot a_{31}\right)\)

\( M_{31}=\left|\begin{bmatrix}a_{12}&\ a_{13}\\a_{22}&\ a_{23}\end{bmatrix}\right|=\left(a_{12}\cdot a_{23}\right)-\left(a_{13}\cdot a_{22}\right)\)

\( M_{32}=\left|\begin{bmatrix}a_{11}&\ a_{13}\\a_{21}&\ a_{23}\end{bmatrix}\right|=\left(a_{11}\cdot a_{23}\right)-\left(a_{13}\cdot a_{21}\right)\)

\( M_{33}=\left|\begin{bmatrix}a_{11}&\ a_{12}\\a_{21}&\ a_{22}\end{bmatrix}\right|=\left(a_{11}\cdot a_{22}\right)-\left(a_{12}\cdot a_{21}\right)\)

where \( M_{ij} \) denotes the minor of the element \( a_{ij} \) of A.

Then the minor of A is \( M=\begin{bmatrix}M_{11}&\ M_{12}&\ M_{13}\\M_{21}&\ M_{22}&\ M_{23}\\M_{31}&\ M_{32}&\ M_{33}\end{bmatrix} \)

Principal Minors of a Matrix

The principal minors of a matrix are the minors of the elements in the principal diagonal, i.e., the elements which have the same row number and column number. Thus when i=j for an element a_ij of a matrix A, the minor found for it after removing the row and column of the same index is called the principal minor of the matrix A.

Let us say we have a matrix, \( A=\begin{bmatrix}a_{12}&\ a_{12}&\ a_{13}\\a_{21}&\ a_{22}&\ a_{23}\\a_{31}&\ a_{32}&\ a_{33}\end{bmatrix} \), then the principal minors will be \( M_{11},\ M_{22},\ M_{33} \), as here the deleted rows and columns have same indices.

Learn about Determinant of 4 x 4 Matrix

Difference between Determinant and Minor of a Matrix

The determinant of a matrix is simply a number that is calculated using all the elements of a given matrix, whereas the minor of a matrix is another matrix formed with the determinants of the smaller order matrix found after eliminating particular rows and columns of the given matrix.

For e.g., if we have a matrix, \( A=\begin{bmatrix}a_{12}&\ a_{12}&\ a_{13}\\a_{21}&\ a_{22}&\ a_{23}\\a_{31}&\ a_{32}&\ a_{33}\end{bmatrix} \), then

Determinant of A, \( \left|A\right|=\left|\begin{matrix}a_{11}&\ a_{12}&\ a_{13}\\a_{21}&\ a_{22}&\ a_{23}\\a_{31}&\ a_{32}&\ a_{33}\end{matrix}\right|=a_{11}\left|\begin{matrix}a_{22}&\ a_{23}\\a_{32}&\ a_{33}\end{matrix}\right|-a_{12}\left|\begin{matrix}a_{21}&\ a_{23}\\a_{31}&\ a_{33}\end{matrix}\right|+a_{13}\left|\begin{matrix}a_{21}&\ a_{22}\\a_{31}&\ a_{32}\end{matrix}\right| \)

\(=a_{11}\left(a_{22}.a_{33}-a_{23}.a_{32}\right)-a_{12}\left(a_{21}.a_{33}-a_{23}.a_{31}\right)+a_{13}\left(a_{21}.a_{32}-a_{22}.a_{31}\right) \)

And if we consider minor of the element \( a_{22} \) of A then

\( M_{22}=\left|\begin{matrix}a_{12}&\ &a_{13}\\a_{31}&\ &a_{33}\end{matrix}\right| \)

and the minor of the matrix A is \( M=\begin{bmatrix}M_{11}&\ M_{12}&\ M_{13}\\M_{21}&\ M_{22}&\ M_{23}\\M_{31}&\ M_{32}&\ M_{33}\end{bmatrix} \)

Thus, the minor of an element of a matrix is the determinant of the newly formed matrix excluding the row and column to which the element belongs and the minor matrix is a matrix formed with these determinants whereas the determinant of a matrix is simply a number calculated considering the whole given matrix.

Learn about important Properties of Determinants

Applications of Minor of a Matrix

The minor of a matrix forms the basis for finding other important features of a matrix with further operations on a matrix. It is used to find the cofactor of a matrix, which can then be used to find the determinant and adjoint of a matrix. The inverse of a matrix is also calculated using the cofactor, determinant and adjoint of a matrix.

Cofactor of Matrix

The cofactor of an element is calculated when we multiply the minor of that element with \( \left(-1\right)^{i+j} \). It is denoted by \( C_{ij} \)

Therefore, \( C_{ij}=\left(-1\right)^{i+j}\cdot M_{ij} \)

Thus for a matrix,

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

we have the cofactor matrix as,

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

Determinant of 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, which as we saw earlier are found using minor elements of the matrix.

If we have a matrix,

\( A=\begin{bmatrix}a_{11}&\ a_{12}&\ a_{13}\\a_{21}&\ a_{22}&\ a_{23}\\a_{31}&\ a_{32}&\ a_{33}\end{bmatrix} \), then the determinant is simply the sum of products of the elements of a row or column and their cofactors, which means

\( \left(C_{11}\times a_{11}\right)+\left(C_{12}\times a_{12}\right)+\left(C_{13}\times a_{13}\right) \), i.e sum of \( C_{ij}\times a_{ij} \) for the elements of a row or column.

Adjoint of a Matrix

We can find the adjoint of a matrix by transposing the cofactor matrix found using the minor elements of the matrix.

Let us a consider a matrix, \( A=\begin{bmatrix}a_{11}&\ a_{12}\\a_{21}&\ a_{22}\end{bmatrix} \), then the cofactor will be,

\( C=\begin{bmatrix}\ \ a_{22}&\ -a_{21}\\-a_{12}&\ \ \ \ \ a_{11}\end{bmatrix} \)

Now to find the adjoint we have to transpose the above matrix.

\( adj\left(A\right)=\begin{bmatrix}\ \ a_{22}&\ -a_{21}\\-a_{12}&\ \ \ \ \ a_{11}\end{bmatrix}^T=\begin{bmatrix}\ \ a_{22}&\ -a_{12}\\-a_{21}&\ \ \ \ \ a_{11}\end{bmatrix} \)

Thus the minor is also used here to find the adjoint of a matrix.

Inverse of a Matrix

We have the formula of the inverse of the matrix as follows.

\( A^{-1}=\frac{1}{\left|A\right|}adj\left(A\right) \) ; \( \left|A\right|\cancel{=}0\)

Here, the adjoint of the matrix and its determinant is found using the minor of the matrix.

Thus the minor of a matrix has vast uses over different matrix operations.

Learn about transformations on a matrix

Row and Column Operations of Determinants

In determinants, certain operations can be performed on rows or columns to simplify calculations. These operations help us solve problems faster, especially when finding the value of a determinant.

(a) Interchanging Rows or Columns:
If we swap the positions of the i-th row and the j-th row (or columns), the determinant changes its sign. This means a positive value becomes negative and vice versa.

(b) Converting a Row to a Column:
Changing a row into a column (or a column into a row) is called transposing the determinant. The value of the determinant remains the same after this.

(c) Multiplying a Row or Column by a Number (k):
If we multiply all elements of a row or column by a number, the value of the determinant also gets multiplied by that number.

(d) Row or Column Addition with Multiplication:
If we multiply one row (or column) by a number k and add it to another row (or column), the determinant value does not change.

Example 1: Find the minor of the given matrix, \( \begin{bmatrix}10&-1\\\ \ 6&\ \ 4\end{bmatrix} \)

Solution: We know that to find the minor of the matrix we have to first find the minor of each element.

So after eliminating the first row and first column, we get the minor of 10 = 4

Similarly, after eliminating respective rows and columns for all elements, we get,

Minor of -1 = 6

Minor of 6 = -1

Minor of 4 = 10

Thus we get the minor of the matrix, \( \begin{bmatrix}\ \ 4\ &\ \ 6\\-1&10\end{bmatrix} \)

Example 2: Find the minor of the element at position a23 in the matrix

[ 1 -9 4 ]

[ 5 6 5 ]

[ -10 2 7 ]

Solution:
The element at position a23 is in the 2nd row and 3rd column.
To find its minor, we eliminate the 2nd row and the 3rd column.

The remaining elements form a 2×2 matrix:

[ 1 -9 ]

[ -10 2 ]

Minor = (1 × 2) − (−9 × −10)

= 2 − 90

= -88

The minor of the element at a23 is −88.

Example 3: Find the minor of the given matrix and hence find its determinant using minors and cofactors.

\( A=\begin{bmatrix}0&2&0\\2&3&4\\4&5&6\end{bmatrix} \)

Solution: First we find the minor of the matrix. So we pick up one element at a time and find the minor.

\(M_{11}=\begin{bmatrix}\cancel{0}&\cancel{2}&\cancel{0}\\\cancel{2}&3&4\\\cancel{4}&5&6\end{bmatrix}=\left|\begin{matrix}3&4\\5&6\end{matrix}\right|=18-20=-2 \)

\( M_{12}=\begin{bmatrix}\cancel{0}&\cancel{2}&\cancel{0}\\2&\cancel{3}&4\\4&\cancel{5}&6\end{bmatrix}=\left|\begin{matrix}2&4\\4&6\end{matrix}\right|=12-16=-4 \)

\( M_{13}=\begin{bmatrix}\cancel{0}&\cancel{2}&\cancel{0}\\2&3&\cancel{4}\\4&5&\cancel{6}\end{bmatrix}=\left|\begin{matrix}2&3\\4&5\end{matrix}\right|=10-12=-2 \)

\( M_{21}=\begin{bmatrix}\cancel{0}&2&0\\\cancel{2}&\cancel{3}&\cancel{4}\\\cancel{4}&5&6\end{bmatrix}=\left|\begin{matrix}2&0\\5&6\end{matrix}\right|=12-0=12 \)

\( M_{22}=\begin{bmatrix}0&\cancel{2}&0\\\cancel{2}&\cancel{3}&\cancel{4}\\4&\cancel{5}&6\end{bmatrix}=\left|\begin{matrix}0&0\\4&6\end{matrix}\right|=0-0=0 \)

\(M_{23}=\begin{bmatrix}0&2&\cancel{0}\\\cancel{2}&\cancel{3}&\cancel{4}\\4&5&\cancel{6}\end{bmatrix}=\left|\begin{matrix}0&2\\4&5\end{matrix}\right|=0-8=-8 \)

\(M_{31}=\begin{bmatrix}\cancel{0}&2&0\\\cancel{2}&3&4\\\cancel{4}&\cancel{5}&\cancel{6}\end{bmatrix}=\left|\begin{matrix}2&0\\3&4\end{matrix}\right|=8-0=8 \)

\(M_{32}=\begin{bmatrix}0&\cancel{2}&0\\2&\cancel{3}&4\\\cancel{4}&\cancel{5}&\cancel{6}\end{bmatrix}=\left|\begin{matrix}0&0\\2&4\end{matrix}\right|=0-0=0 \)

\(M_{33}=\begin{bmatrix}0&2&\cancel{0}\\2&3&\cancel{4}\\\cancel{4}&\cancel{5}&\cancel{6}\end{bmatrix}=\left|\begin{matrix}0&2\\2&3\end{matrix}\right|=0-4=-4 \)

Therefore, we get the minor matrix as,

\( M=\begin{bmatrix}M_{11\ \ }\ M_{12}&M_{13}\\M_{21}\ \ \ M_{22}&M_{23}\\M_{31}\ \ \ M_{32}&M_{33}\end{bmatrix}=\begin{bmatrix}-2&-4&-2\\\ 12&\ \ 0&-8\\\ \ \ 8&\ \ 0&-4\end{bmatrix} \)

Now we know that a determinant is the sum of product of the elements and its cofactor of a row or column. So we consider the second column to find the determinant, and first we find the cofactor of the second column.

\( C_{12}=M_{12}\times\left(-1\right)^{1+2}=-4\times\left(-1\right)^3=4 \)

\( C_{22}=M_{22}\times\left(-1\right)^{2+2}=0\times\left(-1\right)^4=0 \)

\( C_{32}=M_{32}\times\left(-1\right)^{3+2}=0\times\left(-1\right)^5=0 \)

Now we find the determinant,

\(\left|A\right|=\left(C_{12}timesa_{12}\right)+\left(C_{22}timesa_{22}\right)+\left(C_{32}\times a_{32}\right)=\left(4\times2\right)+\left(0\times3\right)+\left(0\times5\right)=8 \)

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