Cofactors in Matrices

1.0What is a Cofactor?

A Cofactor is a key concept in linear algebra that is used to find the determinants, adjoint, and inverse of a matrix.

The cofactor of an element in a square matrix is the determinant of the submatrix that you get by taking out that element's row and column, and multiplying it by a sign factor.

Mathematically, if $A=[a_{ij}]$ is an n×n matrix, then the Cofactor of the element $a_{ij}$ is given by:

$C_{ij}=(-1{)}^{i+j}{M}_{ij}$

Here:

• $M_{ij}$ = minor of the element $a_{ij}$ (determinant of the submatrix obtained after deleting the i-th row and j-th column).
• $(-1{)}^{i+j}=\text{Sign Factor}$

So, cofactors are basically minors with signs attached.

2.0How to Find the Cofactor? 

To find the cofactor of a matrix element, follow these straightforward steps:

Step 1: Select the Element

Choose the element ($a_{ij}$) in the matrix for which you want to find the cofactor.

Step 2: Form the Minor

Remove the i-th row and j-th column containing ($a_{ij}$). The remaining matrix is called the minor matrix of ($a_{ij}$).

Step 3: Calculate the Minor

Find the determinant of the minor matrix. This value is called the minor $((M_{ij}))of(a_{ij})$.

Step 4: Apply the Sign

Multiply the minor by $((-1{)}^{i+j})$ to get the cofactor $((C_{ij})).$

3.0Cofactor Formula 

The standard cofactor formula for a square matrix (A) is: $C_{ij}=(-1{)}^{i+j}\cdot M_{ij}$

Where:

• $(C_{ij})$ = Cofactor of element at (i, j)
• $(M_{ij})$ = Minor of element at (i, j) (determinant of the submatrix formed by removing row i and column j)
• $((-1{)}^{i+j})$ = Sign factor (positive if i+j is even, negative if odd)

4.0Matrix of Cofactors 

The matrix of cofactors is created by replacing every element of the original matrix with its corresponding cofactor. If you have a ($3\times 3$) matrix, your matrix of cofactors will also be ($3\times 3$), but with each element showing the cofactor of the original element in that position.

Why is the Matrix of Cofactors Important?

• To find the adjugate (adjoint) matrix, you take the transpose of the cofactors matrix.
• To find the inverse of a matrix ($(A^{-1})$), you need to use the adjugate matrix.

5.0Cofactors of Determinants 

Cofactors are crucial when you want to expand a determinant. To find the determinant of a square matrix, you can multiply each element in a row (or column) by its cofactor and then add up the results.

Determinant Expansion (using cofactors):

$|A|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}$

or, for a column,

$|A|=a_{1j}{C}_{1j}+a_{2j}{C}_{2j}+\cdots +a_{nj}{C}_{nj}$

This process is called Laplace expansion.

6.0Solved Examples on Cofactors

Below are five step-by-step cofactors examples at the JEE level:

Example 1: Cofactor of an Element in a 2x2 Matrix

Given: $A=\left(\begin{array}{cc}4& 3\\ 2& 5\end{array}\right)$, Find the cofactor of (a_{21}) (element 2).

Solution:

• Remove row 2 and column 1: Left with 3.
• Minor = 3
• Sign = $((-1{)}^{2+1}=-1)$
• Cofactor =$(-1\times 3=-3)$

Example 2: Cofactor in a 3x3 Matrix

Given: $B=\left(\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right)$, Find the cofactor of ($b_{12}$) (element 2).

Solution:

• Remove row 1 and column 2:$\left(\begin{array}{cc}0& 5\\ 1& 6\end{array}\right)$
• Minor = $(0\times 6-1\times 5=-5)$
• Sign = $((-1{)}^{1+2}=-1)$
• Cofactor = $(-1\times -5=5)$

Example 3: Matrix of Cofactors

Given: $C=\left(\begin{array}{cc}2& 1\\ 4& 3\end{array}\right)$, Find the matrix of cofactors.

Solution:

• $(C_{11})$: Remove row 1, col 1: Minor = 3, Sign = +1. Cofactor = 3
• $(C_{12})$: Remove row 1, col 2: Minor = 4, Sign = -1. Cofactor = -4
• $(C_{21})$: Remove row 2, col 1: Minor = 1, Sign = -1. Cofactor = -1
• $(C_{22})$: Remove row 2, col 2: Minor = 2, Sign = +1. Cofactor = 2

Matrix of cofactors: $\left(\begin{array}{cc}3& -4\\ -1& 2\end{array}\right)$

Example 4: Cofactors of Determinants (Expanding a 3x3 Determinant)

Given: $D=\left(\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right)$, Find the determinant by expanding along the first row.

Solution:

• $(C_{11}):Minor=(\left|\begin{array}{cc}1& 4\\ 6& 0\end{array}\right|=(1)(0)-(6)(4)=-24)$, Sign=+1, Cofactor=-24
• $(C_{12}):Minor=(\left|\begin{array}{cc}0& 4\\ 5& 0\end{array}\right|=(0)(0)-(5)(4)=-20)$, Sign=-1, Cofactor=20
• $(C_{13}):Minor=(\left|\begin{array}{cc}0& 1\\ 5& 6\end{array}\right|=(0)(6)-(5)(1)=-5)$, Sign=+1, Cofactor=-5

Expand: $|D|=1\times (-24)+2\times 20+3\times (-5)=-24+40-15=1$

Example 5: Using Cofactors to Find the Inverse

Given: $E=\left(\begin{array}{cc}2& 5\\ 1& 3\end{array}\right)$, Find the inverse using cofactors.

Solution:

1. Find the matrix of cofactors: $(C_{11}=3),(C_{12}=-1),(C_{21}=-5),(C_{22}=2)$
2. Matrix of cofactors: $\left(\begin{array}{cc}3& -1\\ -5& 2\end{array}\right)$
3. Transpose (adjugate):$\left(\begin{array}{cc}3& -5\\ -1& 2\end{array}\right)$
4. Determinant = $(2\times 3-5\times 1=6-5=1)$
5. Inverse = adjugate/determinant: $E^{-1}=\left(\begin{array}{cc}3& -5\\ -1& 2\end{array}\right)$

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