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:

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

The minor of $(a_{11}),(M_{11})$, is: $M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|$

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:

1. Pick the Element: Choose the element ($a_{ij}$) whose minor you need to find.
2. Take out the Row and Column: Remove the column (j) and row (i) that have the element in them.
3. Create the Submatrix: The other elements make up a submatrix of order ( $(n-1)\times (n-1)$ ).
4. 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:

$A=\left[\begin{array}{ccc}2& 3& 1\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

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

• Remove the first row and second column:$\left[\begin{array}{cc}4& 6\\ 7& 9\end{array}\right]$
• 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:$\left[\begin{array}{cc}2& 3\\ 7& 8\end{array}\right]$
• 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:

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

• $\text{Cofactor of }{a}_{ij}=(-1{)}^{i+j}\times M_{ij}$

3. Matrix Inverses: Minors and cofactors are the means for finding the inverse of a matrix using the method of adjoints. 
4. Cramer's Rule: Used to solve systems of linear equations using minors to calculate determinants of submatrices in the solution process. 
5. Eigenvalues and eigenvectors: Minors are preliminary to finding characteristic polynomials for eigenvalues and so minors form part of the process for finding eigenvalues. 
6. 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 $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right]$

Solution:

• $(a_{21}=0)$ (second row, first column)
• Remove second row and first column:$\left[\begin{array}{cc}2& 3\\ 0& 6\end{array}\right]$
• Determinant: $((2\times 6)-(3\times 0)=12-0=12)$
• $(M_{21}=12)$

Example 2: Find all minors in the first row of $B=\left[\begin{array}{ccc}4& 2& 1\\ 3& 5& 6\\ 7& 8& 9\end{array}\right]$

Solution:

$(M_{11})$: Remove first row, first column: $\left[\begin{array}{cc}5& 6\\ 8& 9\end{array}\right]$

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

$(M_{12})$: Remove first row, second column: $\left[\begin{array}{cc}3& 6\\ 7& 9\end{array}\right]$

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

$(M_{13})$: Remove first row, third column: $\left[\begin{array}{cc}3& 5\\ 7& 8\end{array}\right]$

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

Example 3: Find the minor ( $M_{32}$ ) for the matrix $C=\left[\begin{array}{ccc}5& 7& 9\\ 2& 4& 6\\ 1& 3& 8\end{array}\right]$

Solution:

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

Remove the third row and second column: $\left[\begin{array}{cc}5& 9\\ 2& 6\end{array}\right]$

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 $D=\left[\begin{array}{ccc}1& 0& 2\\ -1& 3& 4\\ 2& 5& 6\end{array}\right]$

Solution:

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

Remove the first row and third column: $\left[\begin{array}{cc}-1& 3\\ 2& 5\end{array}\right]$

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 $E=\left[\begin{array}{ccc}3& 8& 1\\ 4& 2& 7\\ 5& 0& 9\end{array}\right]$

Solution:

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

Remove the second row and second column: $\left[\begin{array}{cc}3& 1\\ 5& 9\end{array}\right]$

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

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

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