Adjoint and Inverse of a Matrix

If A and B are non-singular square matrices of the same order such that AB = BA = I (where I is the identity matrix of the same order as A and B), then A and B are called inverses of each other.

We write A⁻¹ = B and B⁻¹ = A.

i.e. AA⁻¹ = A⁻¹A = I.

• If |A| ≠ 0, then A⁻¹ exists.
• If the inverse of a square matrix exists, then it is unique. A matrix can not have more than one distinct inverse.

The adjoint of A is defined as the transpose (i.e. interchange rows and columns) of the cofactor matrix, and it is denoted by adj (A).

For \[ A = [a_{ij}], \]

1. Find the cofactor \[C_{ij}\] of every element.
2. Form the cofactor matrix:
   \[ C = \begin{bmatrix} C_{11} & C_{12} & \cdots \\ C_{21} & C_{22} & \cdots \\ \vdots & \vdots & \end{bmatrix}. \]
3. Transpose the cofactor matrix:
   \[ \boxed{\text{adj}(A) = C^{T}} \]

For \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \]

\[ \boxed{\text{adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}} \]

Shortcut

For a \[2 \times 2\] matrix:

• Interchange the diagonal elements.
• Change the signs of the off-diagonal elements.

• A square matrix \[A\] is singular if \[ \boxed{|A| = 0} \]
  and non-singular if \[ \boxed{|A| \neq 0}. \]
• For a square matrix \[A\] of order \[n\]: \[ \boxed{|\text{adj } A| = |A|^{n-1}} \]
• For non-singular matrices \[A\] and \[B\], \[ \boxed{(AB)^{-1} = B^{-1}A^{-1}} \]
• \[ \boxed{\text{adj}(AB) = (\text{adj } B)(\text{adj } A)} \]
• For every square matrix \[A\] of order \[n\],\[ \boxed{A(\text{adj } A) = (\text{adj } A)A = |A|I} \]
  where \[I\] is the identity matrix of the same order.
• Relation Between Adjoint and Inverse:  A square matrix is invertible if and only if it is non-singular.
  If \[ |A| \neq 0, \] then \[ \boxed{A^{-1} = \frac{1}{|A|}\ \text{adj } A} \]

• Inverse matrices are used in solving systems of equations, which appear in economics, statistics, and engineering applications.

• They help in reversing matrix operations, just as division reverses multiplication in ordinary numbers.

• In computer graphics and data transformations, inverse matrices help recover original positions after rotation or scaling operations.

Analogy:

An inverse matrix works like an “undo” operation for a matrix transformation, provided the matrix is non-singular.

Example: Find the Adjoint

Find \[\text{adj } A\] for \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}. \]

Solution:

Step 1: Find cofactors

\[ C_{11} = 4, \qquad C_{12} = -1 \]

\[ C_{21} = -3, \qquad C_{22} = 2 \]

Step 2: Form the cofactor matrix

\[ C = \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix} \]

Step 3: Take transpose

\[ \text{adj } A = C^{T} \]

\[ \boxed{\text{adj } A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}} \]

 Example: Find \[A^{-1}\] from a Matrix Equation

Solution:

Given \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

and \[ A^2 - 4A + I = O. \]

Step 1: Rearrange

\[ A^2 - 4A = -I \]

\[ A(A - 4I) = -I \]

Step 2: Multiply by \[A^{-1}\]

\[ A - 4I = -A^{-1} \]

Therefore, \[ A^{-1} = 4I - A. \]

Step 3: Substitute \[A\]

\[ A^{-1} = \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} - \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

\[ \boxed{A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}} \]