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Adjoint (Adjugate) Matrix – Complete Guide
The adjoint (or adjugate) matrix of a square matrix plays a central role in inverse matrix computation, determinants, and solving linear systems. This guide explains the definition, formula, worked examples, and practice problems for adjoint matrices.
1. Definition and Formula of Adjoint Matrix
For an \( n \times n \) matrix \( A \), the adjoint matrix, denoted as \( \text{adj}(A) \), is defined as:
\[ \text{adj}(A) = \left( C_{ij} \right)^T \]
Where:- \( C_{ij} \) is the cofactor of element \( a_{ij} \)
- Cofactor is computed as: \[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]
- \( M_{ij} \) is the minor obtained by deleting row \( i \) and column \( j \)
Important property:
\[ A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \quad \text{(if } \det(A) \neq 0) \]
2. Examples
Example 1
Find the adjoint of:
\[ A = \begin{pmatrix} 2 & 4 \\ 3 & 1 \end{pmatrix} \]Show Answer
Step 1: Compute cofactors
\[ C = \begin{pmatrix} (+1)\cdot 1 & -4 \\ -3 & (+1)\cdot 2 \end{pmatrix} = \begin{pmatrix} 1 & -4 \\ -3 & 2 \end{pmatrix} \]Step 2: Transpose cofactor matrix
\[ \text{adj}(A) = \begin{pmatrix} 1 & -3 \\ -4 & 2 \end{pmatrix} \]Example 2
Find the adjoint of:
\[ A = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 4 & 0 \end{pmatrix} \]Show Answer
The cofactor matrix is:
\[ C = \begin{pmatrix} -12 & -2 & 10 \\ 8 & -4 & -4 \\ 6 & 2 & 3 \end{pmatrix} \]Transpose to get adjoint:
\[ \text{adj}(A) = \begin{pmatrix} -12 & 8 & 6 \\ -2 & -4 & 2 \\ 10 & -4 & 3 \end{pmatrix} \]3. Common Mistakes and Tips
- Do not confuse “adjoint” with “transpose” — adjoint uses cofactors.
- Be careful with signs \( (-1)^{i+j} \).
- You must transpose the cofactor matrix at the end.
- If \(\det(A)=0\), adjoint still exists, but \(A^{-1}\) does not.
4. Practice Problems (with Answers)
Exercise 1. Find the adjoint of: \[ A = \begin{pmatrix} 4 & 2 \\ 7 & 1 \end{pmatrix} \]
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\[ \text{adj}(A) = \begin{pmatrix} 1 & -2 \\ -7 & 4 \end{pmatrix} \]Exercise 2. Find the adjoint of: \[ A = \begin{pmatrix} 3 & -1 & 0 \\ 2 & 1 & 4 \\ 0 & 1 & 2 \end{pmatrix} \]
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\[ \text{adj}(A) = \begin{pmatrix} 6 & -2 & 4 \\ -4 & 6 & -12 \\ 2 & -2 & 5 \end{pmatrix} \]Exercise 3. True or False: If \(\det(A) = 0\), then \(\text{adj}(A)\) does not exist.
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False. Adjoint always exists; only the inverse fails to exist.Exercise 4. Use the adjoint to compute the inverse of: \[ A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix} \]
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\[ \text{adj}(A) = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}, \qquad \det(A)=1 \] \[ A^{-1} = \text{adj}(A) \]The adjoint matrix is naturally related to matrix rank, eigenvalues, and triangular form.
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