Inverse of Matrix

Activity - 1

Question: Is there anything like division of matrices?
In the number system, we know a fundamental property:

$$a \times \frac{1}{a} = 1 \quad \text{or} \quad a \cdot a^{-1} = 1$$

Here, $a^{-1}$ is called the multiplicative inverse of $a$.

Key Insight: The product of a number and its inverse is always 1.

Can We Apply This to Matrices?
Do matrix have its 1?

Activity - 2

Let us consider two matrices:

$$A = \begin{bmatrix} 3 & 5 \\ 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} -3 & 5 \\ 2 & -3 \end{bmatrix}$$

Calculate $AB$:

$$AB = \begin{bmatrix} 3 & 5 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} -3 & 5 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} -9+10 & 15-15 \\ -6+6 & 10-9 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Calculate $BA$:

$$BA = \begin{bmatrix} -3 & 5 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} 3 & 5 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} -9+10 & -15+15 \\ 6-6 & 10-9 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Observation:

The matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is the identity matrix.
Acts like the number 1 in matrix operations
$AB = I$ means $B$ "undoes" $A$
$BA = I$ means $A$ "undoes" $B$
Matrices $A$ and $B$ cancle each other out.

Definition of Inverse of a Matrix

Let $A$ and $B$ be two square matrices of the same order. Matrix $B$ is said to be the inverse of matrix $A$ if and only if:

$$AB = BA = I$$

The inverse of $A$ is denoted as:

$$B = A^{-1}$$

Conclusion:

Matrix multipliation is not commutative in general, but if two matrices satisfy $AB = BA$ then one must be inverse of other.

Try this: P = $\begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix}$, Q = $\begin{bmatrix} - 3 & 4 \\ 5 & 1 \end{bmatrix}$

find:

PQ and QP
Does PQ = I?
Does QP = I?
Does PQ = QP?
Are matrices P and Q inverse of each other?

Mathematical Formula for Inverse

Formula for Matrix Inverse

$$A^{-1} = \frac{1}{|A|} \text{Adj}(A)$$

Where:

$|A|$ is the determinant of matrix $A$
$\text{Adj}(A)$ is the adjoint of matrix $A$

Condition for Existence of Inverse

Important:

If $|A| = 0$, then $A^{-1} = \frac{1}{0} \times \text{Adj}(A)$ is undefined
A matrix with determinant $0$ is called a singular matrix
Singular matrices have no inverse
A matrix with determinant $\neq 0$ is called non-singular and has an inverse

The Adjoint of a Matrix

Given:

$$A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$

Step 1: Exchange principal diagonal elements

In the original matrix: $a_{11} = 5$ and $a_{22} = 8$

In the adjoint: $a_{11} = 8$ and $a_{22} = 5$

Step 2: Change signs of secondary diagonal elements

In the original matrix: $a_{12} = 6$ and $a_{21} = 7$

In the adjoint: $a_{12} = -6$ and $a_{21} = -7$

Step 3: Write the adjoint matrix

$$\text{Adj}(A) = \begin{bmatrix} 8 & -6 \\ -7 & 5 \end{bmatrix}$$

✓ For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is simply: $$\text{Adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Step-by-Step Calculation to Find Inverse

Complete Example: Find $A^{-1}$ for the given matrix

Given:

$$A = \begin{bmatrix} 3 & 5 \\ 2 & 3 \end{bmatrix}$$

Step 1: Check if the matrix is singular or non-singular

Find the determinant:

$$|A| = \begin{vmatrix} 3 & 5 \\ 2 & 3 \end{vmatrix} = (3)(3) - (5)(2) = 9 - 10 = -1$$

Since $|A| = -1 \neq 0$, the matrix is non-singular and the inverse exists.

Step 2: Find the adjoint matrix

Using our formula for adjoint of $2 \times 2$ matrix:

$$\text{Adj}(A) = \begin{bmatrix} 3 & -5 \\ -2 & 3 \end{bmatrix}$$

Step 3: Apply the inverse formula

$$A^{-1} = \frac{1}{|A|} \text{Adj}(A) = \frac{1}{-1} \begin{bmatrix} 3 & -5 \\ -2 & 3 \end{bmatrix}$$

$$A^{-1} = \begin{bmatrix} -3 & 5 \\ 2 & -3 \end{bmatrix}$$

Verification: Let's check that $AA^{-1} = I$

$$AA^{-1} = \begin{bmatrix} 3 & 5 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} -3 & 5 \\ 2 & -3 \end{bmatrix}$$ $$= \begin{bmatrix} -9+10 & 15-15 \\ -6+6 & 10-9 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \quad \checkmark$$

Properties of Inverse of a Matrix

There are four important properties of matrix inverses. Let's state and prove each one.

Property I: $(AB)^{-1} = B^{-1}A^{-1}$
Property II: $AA^{-1} = I$
Property III: $|A| = \frac{1}{|A^{-1}|}$ or $|A| \cdot |A^{-1}| = 1$
Property IV: $I^{-1} = I$

Can We Apply This to Matrices?