What is the minor order of a matrix?

The "minor order" of a matrix isn't a standard term; rather, you're likely asking about the minor of an element or the order of a sub-matrix used to find a minor. A minor is the determinant of a sub-matrix

Is cofactor the same as minor?

No, the cofactor and minor are related but not the same. The minor is just the determinant of the submatrix formed by removing a specific row and column.

Can we find minors for a non-square matrix?

No, minors are defined only for square matrices (matrices with the same number of rows and columns).

Why are minors important in matrices?

Minors are essential for calculating determinants, cofactors, and the inverse of a matrix. They are also used in solving linear equations (Cramer’s Rule) and finding eigenvalues.

How many minors does an ( n x n ) matrix have?

An ( n x n ) matrix has ( n^2 ) minors, one for each element of the matrix.

What is the difference between a minor and a determinant?

The determinant is a single value calculated from a square matrix, while a minor is the determinant of a specific submatrix formed by removing one row and one column from the original matrix.

How do minors help in finding the inverse of a matrix?

Minors are used to find cofactors, which are then arranged to form the adjugate (or adjoint) matrix. The inverse of a matrix is given by dividing the adjugate by the determinant of the original matrix.

Are all minors of a matrix always non-zero?

No, minors can be zero or non-zero, depending on the matrix elements and their arrangement.

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Minor of a Matrix

1.0What is Minor of a Matrix?

The determinant of the remaining submatrix after deleting the row and column that contain a given element is known as the minor of that element in a matrix. In order to solve complex mathematical problems, such as those found in the JEE syllabus, it is essential to comprehend determinants, cofactors, and the inverse of matrices.

2.0Minor of a Matrix Formula

The minor of a matrix formula is quite straightforward: $[M_{ij} = det (A_{ij})]$

Where ( $M_{ij}$ ) is the minor of the element in the ( i )-th row and ( j )-th column, and ( $A_{ij}$ ) is the submatrix formed by deleting the ( i )-th row and ( j )-th column from matrix ( A ).

For a 3×3 matrix:

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33}$

The minor of $(a_{11}), (M_{11})$ , is: $M_{11} = a_{22} a_{32} a_{23} a_{33}$

3.0How to Find the Minor of a Matrix?

If you want to know how to find the minor of a matrix, just do these things:

Pick the Element: Choose the element ( $a_{ij}$ ) whose minor you need to find.
Take out the Row and Column: Remove the column (j) and row (i) that have the element in them.
Create the Submatrix: The other elements make up a submatrix of order ( $(n - 1) \times (n - 1)$ ).
Find the Determinant: This value is the required minor; find the determinant of this submatrix ( $M_{ij}$ ).

Tip: Always make sure you're removing the right row and column, because even a small mistake can change the answer completely.

4.0Minors of a 3×3 Matrix

Let’s take a 3×3 matrix for better understanding:

$A = 247358169$

Find the Minor of ( $a_{12}$ ) (Element 3):

Remove the first row and second column: $[4769]$
Calculate its determinant: $((4 \times 9) - (6 \times 7) = 36 - 42 = - 6)$
So, $(M_{12} = - 6)$

Find the Minor of ( $a_{23}$ ) (Element 6):

Remove the second row and third column: $[2738]$
Determinant: $((2 \times 8) - (3 \times 7) = 16 - 21 = - 5)$
So, $(M_{23} = - 5)$

In general, the minor of an element in a 3×3 matrix is the determinant of the 2×2 submatrix that is left after deleting its row and column.

5.0Applications of Minor of Matrix

Understanding the applications of minors of a matrix is important, especially for students preparing for competitive exams like JEE. Some significant applications are:

Determinate Expansion: Minors are used in the Laplace expansion when calculating the determinant of higher order matrices.
Finding Cofactors: The cofactor of an element is determined by its minor, which is used for determinant expansion or finding the adjoint.

$Cofactor of a_{ij} = (- 1)^{i + j} \times M_{ij}$

Matrix Inverses: Minors and cofactors are the means for finding the inverse of a matrix using the method of adjoints.
Cramer's Rule: Used to solve systems of linear equations using minors to calculate determinants of submatrices in the solution process.
Eigenvalues and eigenvectors: Minors are preliminary to finding characteristic polynomials for eigenvalues and so minors form part of the process for finding eigenvalues.
Engineering and Physics: Minors are used in analyzing electrical circuits, mechanics and structural engineering problems.

6.0Solved Examples on Minor of Matrix

Example 1: Find the minor ( $M_{21}$ ) for the matrix $A = 101240356$

Solution:

$(a_{21} = 0)$ (second row, first column)
Remove second row and first column: $[2036]$
Determinant: $((2 \times 6) - (3 \times 0) = 12 - 0 = 12)$
$(M_{21} = 12)$

Example 2: Find all minors in the first row of $B = 437258169$

Solution:

$(M_{11})$ : Remove first row, first column: $[5869]$

$(= (5 \times 9) - (6 \times 8) = 45 - 48 = - 3)$

$(M_{12})$ : Remove first row, second column: $[3769]$

$(= (3 \times 9) - (6 \times 7) = 27 - 42 = - 15)$

$(M_{13})$ : Remove first row, third column: $[3758]$

$(= (3 \times 8) - (5 \times 7) = 24 - 35 = - 11)$

Example 3: Find the minor ( $M_{32}$ ) for the matrix $C = 521743968$

Solution:

( $a_{32} = 3$ ) (third row, second column)

Remove the third row and second column: $[5296]$

Find the determinant: $(5 \times 6) - (9 \times 2) = 30 - 18 = 12$

Therefore, $(M_{32} = 12) .$

Example 4: Find the minor ( $M_{13}$ ) for the matrix $D = 1 - 1 2 035246$

Solution:

( $a_{13} = 2$ ) (first row, third column)

Remove the first row and third column: $[- 1 2 35]$

Find the determinant: $(- 1 \times 5) - (3 \times 2) = - 5 - 6 = - 11$

Therefore, ( $M_{13} = - 11$ ).

Example 5: Find the minor ( $M_{22}$ ) for the matrix $E = 345820179$

Solution:

( $a_{22} = 2$ ) (second row, second column)

Remove the second row and second column: $[3519]$

Find the determinant: $[(3 \times 9) - (1 \times 5) = 27 - 5 = 22]$

Therefore, $(M_{22} = 22) .$

Also Read:

Matrices	Adjoint of a Matrix
Matrix Operations	Determinants and Matrices

Frequently Asked Questions

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Name

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Class

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Goal

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Preferred Programs

Preferred Mode

State

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I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Minor of a Matrix

1.0What is Minor of a Matrix?

The determinant of the remaining submatrix after deleting the row and column that contain a given element is known as the minor of that element in a matrix. In order to solve complex mathematical problems, such as those found in the JEE syllabus, it is essential to comprehend determinants, cofactors, and the inverse of matrices.

2.0Minor of a Matrix Formula

The minor of a matrix formula is quite straightforward: $[M_{ij} = det (A_{ij})]$

Where ( $M_{ij}$ ) is the minor of the element in the ( i )-th row and ( j )-th column, and ( $A_{ij}$ ) is the submatrix formed by deleting the ( i )-th row and ( j )-th column from matrix ( A ).

For a 3×3 matrix:

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33}$

The minor of $(a_{11}), (M_{11})$ , is: $M_{11} = a_{22} a_{32} a_{23} a_{33}$

3.0How to Find the Minor of a Matrix?

If you want to know how to find the minor of a matrix, just do these things:

Pick the Element: Choose the element ( $a_{ij}$ ) whose minor you need to find.
Take out the Row and Column: Remove the column (j) and row (i) that have the element in them.
Create the Submatrix: The other elements make up a submatrix of order ( $(n - 1) \times (n - 1)$ ).
Find the Determinant: This value is the required minor; find the determinant of this submatrix ( $M_{ij}$ ).

Tip: Always make sure you're removing the right row and column, because even a small mistake can change the answer completely.

4.0Minors of a 3×3 Matrix

Let’s take a 3×3 matrix for better understanding:

$A = 247358169$

Find the Minor of ( $a_{12}$ ) (Element 3):

Remove the first row and second column: $[4769]$
Calculate its determinant: $((4 \times 9) - (6 \times 7) = 36 - 42 = - 6)$
So, $(M_{12} = - 6)$

Find the Minor of ( $a_{23}$ ) (Element 6):

Remove the second row and third column: $[2738]$
Determinant: $((2 \times 8) - (3 \times 7) = 16 - 21 = - 5)$
So, $(M_{23} = - 5)$

In general, the minor of an element in a 3×3 matrix is the determinant of the 2×2 submatrix that is left after deleting its row and column.

5.0Applications of Minor of Matrix

Understanding the applications of minors of a matrix is important, especially for students preparing for competitive exams like JEE. Some significant applications are:

Determinate Expansion: Minors are used in the Laplace expansion when calculating the determinant of higher order matrices.
Finding Cofactors: The cofactor of an element is determined by its minor, which is used for determinant expansion or finding the adjoint.

$Cofactor of a_{ij} = (- 1)^{i + j} \times M_{ij}$

Matrix Inverses: Minors and cofactors are the means for finding the inverse of a matrix using the method of adjoints.
Cramer's Rule: Used to solve systems of linear equations using minors to calculate determinants of submatrices in the solution process.
Eigenvalues and eigenvectors: Minors are preliminary to finding characteristic polynomials for eigenvalues and so minors form part of the process for finding eigenvalues.
Engineering and Physics: Minors are used in analyzing electrical circuits, mechanics and structural engineering problems.

6.0Solved Examples on Minor of Matrix

Example 1: Find the minor ( $M_{21}$ ) for the matrix $A = 101240356$

Solution:

$(a_{21} = 0)$ (second row, first column)
Remove second row and first column: $[2036]$
Determinant: $((2 \times 6) - (3 \times 0) = 12 - 0 = 12)$
$(M_{21} = 12)$

Example 2: Find all minors in the first row of $B = 437258169$

Solution:

$(M_{11})$ : Remove first row, first column: $[5869]$

$(= (5 \times 9) - (6 \times 8) = 45 - 48 = - 3)$

$(M_{12})$ : Remove first row, second column: $[3769]$

$(= (3 \times 9) - (6 \times 7) = 27 - 42 = - 15)$

$(M_{13})$ : Remove first row, third column: $[3758]$

$(= (3 \times 8) - (5 \times 7) = 24 - 35 = - 11)$

Example 3: Find the minor ( $M_{32}$ ) for the matrix $C = 521743968$

Solution:

( $a_{32} = 3$ ) (third row, second column)

Remove the third row and second column: $[5296]$

Find the determinant: $(5 \times 6) - (9 \times 2) = 30 - 18 = 12$

Therefore, $(M_{32} = 12) .$

Example 4: Find the minor ( $M_{13}$ ) for the matrix $D = 1 - 1 2 035246$

Solution:

( $a_{13} = 2$ ) (first row, third column)

Remove the first row and third column: $[- 1 2 35]$

Find the determinant: $(- 1 \times 5) - (3 \times 2) = - 5 - 6 = - 11$

Therefore, ( $M_{13} = - 11$ ).

Example 5: Find the minor ( $M_{22}$ ) for the matrix $E = 345820179$

Solution:

( $a_{22} = 2$ ) (second row, second column)

Remove the second row and second column: $[3519]$

Find the determinant: $[(3 \times 9) - (1 \times 5) = 27 - 5 = 22]$

Therefore, $(M_{22} = 22) .$

Also Read:

Matrices	Adjoint of a Matrix
Matrix Operations	Determinants and Matrices

Frequently Asked Questions

What is the minor order of a matrix?

Is cofactor the same as minor?

Can we find minors for a non-square matrix?

Why are minors important in matrices?

How many minors does an ( n x n ) matrix have?

What is the difference between a minor and a determinant?

How do minors help in finding the inverse of a matrix?

Are all minors of a matrix always non-zero?

Join ALLEN!

Minor of a Matrix

1.0What is Minor of a Matrix?

2.0Minor of a Matrix Formula

3.0How to Find the Minor of a Matrix?

4.0Minors of a 3×3 Matrix

5.0Applications of Minor of Matrix

6.0Solved Examples on Minor of Matrix

Table of Contents

Frequently Asked Questions

What is the minor order of a matrix?

Is cofactor the same as minor?

Can we find minors for a non-square matrix?

Why are minors important in matrices?

How many minors does an ( n x n ) matrix have?

What is the difference between a minor and a determinant?

How do minors help in finding the inverse of a matrix?

Are all minors of a matrix always non-zero?

Join ALLEN!

Minor of a Matrix

1.0What is Minor of a Matrix?

2.0Minor of a Matrix Formula

3.0How to Find the Minor of a Matrix?

4.0Minors of a 3×3 Matrix

5.0Applications of Minor of Matrix

6.0Solved Examples on Minor of Matrix

Table of Contents