# [[Cofactor Expansion and Matrix Inverse]] (Mathematics > Linear Algebra) ## Definition The **cofactor** of an element $a_{ij}$ in a square matrix $A$ is defined as: $C_{ij} = (-1)^{i+j} \det(M_{ij}),$ where $M_{ij}$ is the minor of $a_{ij}$ (the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column of $A$). The **inverse** of a matrix $A$ is given by: $A^{-1} = \frac{1}{\det(A)} C^T,$ where $C^T$ is the transpose of the cofactor matrix of $A$. ## Key Concepts - **Minor**: The determinant of a smaller matrix obtained by removing a row and a column from $A$. - **Cofactor**: Includes the sign factor $(-1)^{i+j}$ to account for the position in $A$. - **Cofactor Matrix**: A matrix where each element is the cofactor $C_{ij}$ of the corresponding element $a_{ij}$ in $A$. - **Adjugate**: The transpose of the cofactor matrix. ## Important Properties 1. The cofactor expansion can compute the determinant: $\det(A) = \sum_{j=1}^n a_{ij} C_{ij}, \quad \text{for any row } i.$ 2. The inverse exists only if $\det(A) \neq 0$. 3. The inverse formula: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A),$ where $\text{adj}(A)$ is the adjugate matrix. ## Essential Steps to Find $A^{-1}$ 1. Compute $\det(A)$: - Use cofactor expansion along any row or column. - If $\det(A) = 0$, $A$ is not invertible. 2. Calculate cofactors $C_{ij}$ for each element of $A$. 3. Form the cofactor matrix. 4. Transpose the cofactor matrix to get $\text{adj}(A)$. 5. Use the formula: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A).$ ## Core Examples 1. **Example 1**: Find $a'_{31}$ in $A^{-1}$ for: $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}.$ - Compute $C_{13}$: $C_{13} = (-1)^{1+3} \det\begin{bmatrix} 0 & 1 \\ 5 & 6 \end{bmatrix} = (-1)^4 (0 \cdot 6 - 1 \cdot 5) = -5.$ - Compute $\det(A)$: $\det(A) = 1\cdot(1\cdot0 - 4\cdot6) - 2\cdot(0\cdot0 - 4\cdot5) + 3\cdot(0\cdot6 - 1\cdot5) = -24 + 40 - 15 = 1.$ - Entry $a'_{31}$ in $A^{-1}$: $a'_{31} = \frac{1}{\det(A)}C_{13} = \frac{1}{1}(-5) = -5.$ 2. **Example 2**: For $A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$: - Compute $\det(A) = 2\cdot3 - 1\cdot5 = 6 - 5 = 1.$ - Compute cofactors: $C_{11} = 3, \, C_{12} = -5, \, C_{21} = -1, \, C_{22} = 2.$ - Adjugate matrix: $\text{adj}(A) = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.$ - Inverse: $A^{-1} = \frac{1}{\det(A)}\text{adj}(A) = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.$ ## Related Theorems/Rules - [[Determinants and Invertibility]] - [[Row Reduction for Inverses]] ## Common Pitfalls - Mistaking minors for cofactors (forgetting $(-1)^{i+j}$). - Incorrectly calculating the determinant or minor. - Using the formula without verifying $\det(A) \neq 0$. ## Related Topics - [[Determinant of a 3x3 Matrix]] - [[Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method]] - [[Minor of a Matrix Element]] - [[Matrix Multiplication]] - [[Singular Matrices]] - [[Matrix Inversion]] - [[Special Types of Matrices]] - [[Inverse of a 2×2 Matrix]] - [[Multiplicative Inverse]] - [[Identity Matrix]] - [[Determinants and Cofactor Expansion]] - [[Matrix Inverses]] - [[Adjugate Matrix]] ## Quick Review Questions 1. Compute the cofactor $C_{22}$ for $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$. 2. What is the formula for $A^{-1}$ in terms of $\det(A)$ and the adjugate matrix? 3. Verify the inverse of $A = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$ using cofactors. *** ![[Cofactor_Expansion_and_Matrix_Inverse_visualization]]