Inverse Matrices and Adjugate Matrices

Inverse Matrices

The inverse matrix is an important concept in matrix theory, with important applications in solving systems of linear equations, matrix decompositions and so on.

Note:

• Only square matrices can possibly have an inverse
• If the matrix $A$ is invertible, its inverse is unique
• Not every square matrix has an inverse

Properties of the Inverse Matrix

1. Uniqueness: if $A$ is invertible, its inverse is unique
2. Double inverse: $(A^{-1})^{-1} = A$
3. Inverse of a product: $(AB)^{-1} = B^{-1}A^{-1}$ (if both $A$ and $B$ are invertible)
4. Inverse of a transpose: $(A^T)^{-1} = (A^{-1})^T$
5. Inverse of a scalar multiple: $(kA)^{-1} = \frac{1}{k}A^{-1}$ ($k \neq 0$)

Methods for Computing the Inverse

Method 1: The Adjugate Matrix Method

Example: Find the inverse of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Solution:

1. $|A| = 1 \times 4 - 2 \times 3 = 4 - 6 = -2 \neq 0$, so $A$ is invertible
2. $A_{11} = 4$, $A_{12} = -3$, $A_{21} = -2$, $A_{22} = 1$
3. $A^* = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$
4. $A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$

Method 2: The Elementary Transformation Method

Steps:

1. Form the augmented matrix $[A|I]$
2. Apply elementary row operations to the augmented matrix to reduce $A$ to the identity matrix
3. At that point the position of $I$ is $A^{-1}$

Example: Find the inverse of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ by the elementary transformation method.

Solution: $\begin{pmatrix} 1 & 2 & | & 1 & 0 \\ 3 & 4 & | & 0 & 1 \end{pmatrix}$

$R_2 - 3R_1$: $\begin{pmatrix} 1 & 2 & | & 1 & 0 \\ 0 & -2 & | & -3 & 1 \end{pmatrix}$

$R_2 \times (-\frac{1}{2})$: $\begin{pmatrix} 1 & 2 & | & 1 & 0 \\ 0 & 1 & | & 1.5 & -0.5 \end{pmatrix}$

$R_1 - 2R_2$: $\begin{pmatrix} 1 & 0 & | & -2 & 1 \\ 0 & 1 & | & 1.5 & -0.5 \end{pmatrix}$

so $A^{-1} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$

Adjugate Matrices

The adjugate matrix is an important tool for computing the inverse matrix, and it provides an explicit formula for it.

The Definition of the Adjugate Matrix

Let $A = (a_{ij})_{n \times n}$ be an n-th order square matrix, and let $A_{ij}$ be the algebraic cofactor of the entry $a_{ij}$. Then the matrix

$$A^* = \begin{pmatrix} A_{11} & A_{21} & \cdots & A_{n1} \\ A_{12} & A_{22} & \cdots & A_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ A_{1n} & A_{2n} & \cdots & A_{nn} \end{pmatrix}$$

is called the adjugate matrix of $A$.

Note: the entry in row $i$, column $j$ of the adjugate matrix is $A_{ji}$, that is, the transpose of the algebraic cofactors.

Properties of the Adjugate Matrix

1. Basic relation: $AA^* = A^*A = |A|I$
2. Inverse matrix formula: if $|A| \neq 0$, then $A^{-1} = \frac{1}{|A|}A^*$
3. Transpose property: $(A^*)^T = (A^T)^*$
4. Scalar multiple property: $(kA)^* = k^{n-1}A^*$

Computing the Adjugate Matrix

Steps:

1. Compute the algebraic cofactor of each entry
2. Arrange them in the transposed positions

Example: Find the adjugate matrix of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$.

Solution:

1. Compute the algebraic cofactor of each entry:

   • $A_{11} = \begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} = -24$
   • $A_{12} = -\begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} = 20$
   • $A_{13} = \begin{vmatrix} 0 & 1 \\ 5 & 6 \end{vmatrix} = -5$
   • $A_{21} = -\begin{vmatrix} 2 & 3 \\ 6 & 0 \end{vmatrix} = 18$
   • $A_{22} = \begin{vmatrix} 1 & 3 \\ 5 & 0 \end{vmatrix} = -15$
   • $A_{23} = -\begin{vmatrix} 1 & 2 \\ 5 & 6 \end{vmatrix} = 4$
   • $A_{31} = \begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} = 5$
   • $A_{32} = -\begin{vmatrix} 1 & 3 \\ 0 & 4 \end{vmatrix} = -4$
   • $A_{33} = \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} = 1$

2. Arrange them in the transposed positions: $\text{adj}(A) = \begin{pmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{pmatrix}$

Summary

Symbols Used in This Article