Adjoint of a Matrix

The adjoint of a matrix is a fundamental concept in linear algebra, closely related to the inverse of a matrix. It is a matrix composed of the cofactors of the original matrix, transposed. This concept is particularly important in solving systems of linear equations, finding inverses, and other matrix-related computations.

1.0What is the Adjoint of a Matrix?

The adjoint of a matrix (sometimes called the adjugate) is the transpose of the cofactor matrix of a given square matrix. For a matrix A, the adjoint is denoted by adj(A). The cofactor matrix is created by replacing each element of A with its corresponding cofactor.

2.0Adjoint of a Matrix Formula

Given a square matrix A, the adjoint of A, denoted by adj(A), is the transpose of its cofactor matrix.

Here's the step-by-step process to compute the adjoint:

1. Cofactor Matrix: For each element a_ij​ of the matrix A, determine its cofactor C_ij​. The cofactor is the determinant of the minor matrix that remains after removing the i^th row and j^th column, multiplied by (−1)^i+j.
2. Construct the Cofactor Matrix: Arrange all the cofactors C_ij​​ into a matrix, called the cofactor matrix.
3. Transpose the Cofactor Matrix: Swap the rows and columns of the cofactor matrix to obtain the adjoint of the matrix A.

Mathematically, if A is a n × n matrix, then the adjoint of A, denoted by adj(A), is given by:

$adj(A)=Cofactor(A^T)$

3.0Mathematical Representation

If A is a square matrix of order n, then the adjoint of A, adj(A), is given by:

$adj(A)=\left[\begin{array}{cccc}{C}_{11}& C_{21}& ...& C_{n1}\\ C_{12}& C_{22}& ...& C_{n2}\\ ...& ...& ...& ...\\ C_{1n}& C_{2n}& ....& C_{nn}\end{array}\right]$

Where C_ij is the cofactor of the element a_ij in matrix A.

4.0Adjoint of a 2 × 2 Matrix

For a 2 × 2 matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ , the adjoint is computed as follows:

1. Find the Cofactors:

• C₁₁ = d 
• C₁₂ = –c 
• C₂₁ = –b 
• C₂₂ = a 

2. Construct the Cofactor Matrix:

$\text{Cofactor(A)}=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

3. Transpose the Cofactor Matrix:

$\text{adj(A)}=\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]$

So, the adjoint of matrix A is: $\text{adj(A)}=\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]$

5.0Adjoint of a 3 × 3 Matrix

Consider the matrix A as: $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right]$

1. Find the Cofactors

• Cofactor of A11: $\text{Minor of}{A}_{11}=\left|\begin{array}{cc}4& 5\\ 0& 6\end{array}\right|=4(6)-5(0)=24$
• Cofactor of A₁₂: $\text{Minor of}{A}_{12}=\left|\begin{array}{cc}0& 5\\ 1& 6\end{array}\right|=0(6)-5(1)=-5$

2. Construct the Cofactor Matrix

$Cofactor(A)=\left[\begin{array}{ccc}24& 5& -4\\ -12& 3& 2\\ -2& -5& 4\end{array}\right]$

3. Transpose the Cofactor Matrix

$adj(A)=\left[\begin{array}{ccc}24& -12& -2\\ 5& 3& -5\\ -4& 2& 4\end{array}\right]$

Thus, the adjoint of the matrix A is:

$adj(A)=\left[\begin{array}{ccc}24& -12& 0\\ 5& 3& -15\\ -4& 2& 4\end{array}\right]$

6.0Cofactor and Adjoint of a Matrix

The concept of the cofactor is central to finding the adjoint of a matrix. For any square matrix A, the cofactor of an element a_ij (where i is the row and j is the column) is the determinant of the submatrix that remains after removing the i^th row and j^th column, multiplied by (–1)^i+j.

7.0Importance of the Adjoint of a Matrix

The adjoint of a matrix is crucial in finding the inverse of a matrix. For a non-singular matrix A, the inverse is given by: $A^{-1}=\frac{1}{det(A)}adj(A)$

Where det(A) is the determinant of A. If the determinant is non-zero, the inverse exists, and the adjoint is used to compute it.

8.0Adjoint of a Matrix Example

Example 1:  Let’s find the adjoint of a 2 × 2 matrix B:

$B=\left[\begin{array}{cc}2& 1\\ 5& 3\end{array}\right]$

Solution: 

1. Find the Cofactors

• Cofactor of B₁₁: Cofactor B₁₁ = 3 
• Cofactor of B₁₂: Cofactor B₁₂ = –5 
• Cofactor of B₂₁: Cofactor B₂₁ = –1 
• Cofactor of B₂₂: Cofactor B₂₂= 2 

2. Construct the Cofactor Matrix

$Cofactor(B)=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$

3. Transpose the Cofactor Matrix

$adj(B)=\left[\begin{array}{cc}3& -1\\ -5& 2\end{array}\right]$

So, the adjoint of matrix B is:

$adj(B)=\left[\begin{array}{cc}3& -1\\ -5& 2\end{array}\right]$

Example 2: Find the Adjoint of a 3 × 3 Matrix. Given the matrix A:

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

Solution: 

1. Find the Cofactors

To find the cofactor of each element in the matrix, we need to determine the determinant of the 2 × 2 submatrices that remain after removing the respective row and column of each element.

Cofactor of a₁₁ = 2:

 $\text{Minor of}{a}_{11}=\left|\begin{array}{cc}1& -3\\ 2& 5\end{array}\right|$ $=(1)(5)–(–3)(2)=5+6=11$

Cofactor C₁₁ = 11.

Cofactor of a₁₂ = 3:

Minor of a₁₂ = $\left|\begin{array}{cc}4& -3\\ -1& 5\end{array}\right|$ $=(4)(5)–(–3)(–1)$ $=20–3=17$

Cofactor C₁₂ = –17 (note the sign change due to the position of the element).

Cofactor of a₁₃ = 1:

Minor of a₁₃ = $\left|\begin{array}{cc}4& 1\\ -1& 2\end{array}\right|$$=(4)(2)–(1)(–1)=8+1=9$

Cofactor C₁₃ = 9.

Cofactor of a₂₁ = 4:

Minor of a₂₁ = $\left|\begin{array}{cc}3& 1\\ 2& 5\end{array}\right|$ $=(3)(5)–(1)(2)=15–2=13$

Cofactor C₂₁ = –13 (note the sign change due to the position of the element).

Cofactor of a₂₂ = 1:

Minor of a₂₂ = $\left|\begin{array}{cc}2& 1\\ -1& 5\end{array}\right|$ $=(2)(5)–(1)(–1)=10+1=11$

Cofactor C₂₂ = 11.

Cofactor of a₂₃ = –3:

Minor of a₂₃ = $\left|\begin{array}{cc}2& 3\\ -1& 2\end{array}\right|$$=(2)(2)–(3)(–1)=4+3=7$

Cofactor C₂₃ = –7 (note the sign change due to the position of the element).

Cofactor of a₃₁ = –1:

Minor of a₃₁ = $\left|\begin{array}{cc}3& 1\\ 1& -3\end{array}\right|$$=(3)(–3)–(1)(1)=–9–1=–10$

Cofactor C₃₁ = –10.

Cofactor of a₃₂ = 2:

Minor of a₃₂ = $\left|\begin{array}{cc}2& 1\\ 4& -3\end{array}\right|$$=(2)(–3)–(1)(4)=–6–4=–10$

Cofactor C₃₂ = 10 (note the sign change due to the position of the element).

Cofactor of a₃₃ = 5:

Minor of a₃₃ =$\left|\begin{array}{cc}2& 3\\ 4& 1\end{array}\right|$$=(2)(1)–(3)(4)=2–12=–10$

Cofactor C₃₃ = –10.

2. Construct the Cofactor Matrix

Now, arrange all the cofactors into a cofactor matrix:

$Cofactor(A)=\left[\begin{array}{ccc}11& -17& 9\\ -13& 11& -7\\ -10& 10& -10\end{array}\right]$

3. Transpose the Cofactor Matrix

Finally, transpose the cofactor matrix to get the adjoint of the matrix A:

$adj(A)=\left[\begin{array}{ccc}11& -13& -10\\ -17& 11& 10\\ 9& -7& -10\end{array}\right]$

The adjoint of the matrix A is:

$adj(A)=\left[\begin{array}{ccc}11& -13& -10\\ -17& 11& 10\\ 9& -7& -10\end{array}\right]$

Example 3: Adjoint of a 2 × 2 Matrix. Consider the matrix B: $B=\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$

Solutions:

1. Find the Cofactors

Cofactor of b₁₁ = 4:

Minor of b₁₁ = 6

Cofactor C₁₁ = 6 .

Cofactor of b₁₂ = 7:

Minor of b₁₂ = 2

Cofactor C₁₂ = –2. (since it’s in an odd position).

Cofactor of b₂₁ = 2:

Minor of b₂₁ = 7

Cofactor C₂₁ = –7. (since it’s in an odd position).

Cofactor of b₂₂ = 6:

Minor of b₂₂ = 4

Cofactor C₂₂ = 4.

2. Construct the Cofactor Matrix

$Cofactor(B)=\left[\begin{array}{cc}6& -2\\ -7& 4\end{array}\right]$

3. Transpose the Cofactor Matrix

$adj(B)=\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]$

Final Answer

The adjoint of matrix B is:

$adj(B)=\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]$

Example 4: Adjoint of a 3x3 Matrix. Consider the matrix C:

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

Solutions:

1. Find the Cofactors

Cofactor of c₁₁ = 2:

Minor of c₁₁ = $\left|\begin{array}{cc}4& 5\\ 0& 6\end{array}\right|$$=(4)(6)–(5)(0)=24$

Cofactor C₁₁ = 24.

Cofactor of c₁₂ = 1:

Minor of c₁₂ =$\left|\begin{array}{cc}0& 5\\ 1& 6\end{array}\right|$$=(0)(6)–(5)(1)=–5$

Cofactor C₁₂ = 5. (since it’s in an odd position).

Cofactor of c₁₃ = 3:

Minor of c₁₃ =$\left|\begin{array}{cc}0& 4\\ 1& 0\end{array}\right|$$=(0)(0)–(4)(1)=–4$

Cofactor C₁₃ = –4.

Cofactor of c₂₁ = 0:

Minor of c₂₁ = $\left|\begin{array}{cc}1& 3\\ 0& 6\end{array}\right|$$=(1)(6)–(3)(0)=6$

Cofactor C₂₁ = –6. (since it’s in an odd position).

Cofactor of c₂₂ = 4:

Minor of c₂₂ = $\left|\begin{array}{cc}2& 3\\ 1& 6\end{array}\right|$$=(2)(6)–(3)(1)=12–3=9$

Cofactor C₂₂ = 9.

Cofactor of c₂₃ = 5:

Minor of c₂₃ = $\left|\begin{array}{cc}2& 1\\ 1& 0\end{array}\right|$$=(2)(0)–(1)(1)=–1$

Cofactor C₂₃ = 1. (since it’s in an odd position).

Cofactor of c₃₁ = 1:

Minor of c₃₁ = $\left|\begin{array}{cc}1& 3\\ 4& 5\end{array}\right|$$=(1)(5)–(3)(4)=5–12=–7$

Cofactor C₃₁ = –7.

Cofactor of c₃₂ = 0:

Minor of c₃₂ = $\left|\begin{array}{cc}2& 3\\ 1& 5\end{array}\right|$$=(2)(5)-(3)(1)=10-3=7$

Cofactor C₃₂ = –7. (since it’s in an odd position).

Cofactor of c₃₃ = 6:

Minor of c₃₃ = $\left|\begin{array}{cc}2& 1\\ 0& 4\end{array}\right|$$=(2)(4)-(1)(0)=8$

Cofactor C₃₃ = 8.

2. Construct the Cofactor Matrix

$Cofactor(C)=\left[\begin{array}{ccc}24& 5& -4\\ -6& 9& -1\\ -7& 10& 8\end{array}\right]$

3. Transpose the Cofactor Matrix

$adj(C)=\left[\begin{array}{ccc}24& 6& -7\\ 5& 9& 10\\ -4& -1& 8\end{array}\right]$

Final Answer

The adjoint of matrix C is:

$adj(C)=\left[\begin{array}{ccc}24& 6& -7\\ 5& 9& 10\\ -4& -1& 8\end{array}\right]$

Example 5: Consider matrix D and find the adjoint of matrix D.

$D=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Solutions:

1. Find the Cofactors

Cofactor of d₁₁ = 0:

Minor of d₁₁ = 0

Cofactor C₁₁ = 0.

Cofactor of d₁₂ = 1:

Minor of d₁₂ = –1

Cofactor C₁₂ = 1. (since it’s in an odd position).

Cofactor of d₂₁ = –1:

Minor of d₂₁ = 1

Cofactor C₂₁ = –1. (since it’s in an odd position).

Cofactor of d₂₂ = 0:

Minor of d₂₂ = 0

Cofactor C₂₂ = 0.

2. Construct the Cofactor Matrix

$Cofactor(D)=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

3. Transpose the Cofactor Matrix

$adj(D)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Final Answer

The adjoint of matrix D is: $adj(D)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

9.0Practice Problems On Adjoint of a Matrix Questions

1. Find the adjoint of the following matrix: $A=\left[\begin{array}{ccc}4& 7& 2\\ 3& 6& 1\\ -2& -5& 0\end{array}\right]$
2. Find the adjoint of the following matrix: $B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$
3. Given a 2 × 2 matrix $C=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ , find adj(C).
4. Calculate the adjoint of a 4 × 4 matrix D:$D=\left[\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\\ 3& 4& 1& 2\\ 4& 1& 2& 3\end{array}\right]$