🟣 Matrix Algebra: Determinants & Inverses

The determinant is a special number that can be calculated from a square matrix, offering insights into the matrix's properties. The inverse of a matrix, similar to the reciprocal of a number, allows us to 'divide' in matrix algebra, in a sense. In this part, we will focus on $2 \times 2$ matrices to build a solid foundation.

For a square matrix, the Determinant is a special number that can be calculated from its elements. For a $2 \times 2$ matrix, it's a simple calculation. The Inverse of a square matrix (if it exists) is another matrix that, when multiplied by the original matrix, gives the Identity Matrix.

For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is defined as follows:

The determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, denoted as $\det(A)$ or $|A|$, is given by:

$$\det(A) = ad - bc$$

It's the product of the main diagonal elements ($a$ and $d$) minus the product of the off-diagonal elements ($b$ and $c$).

Example 1: Determinant of a 2×2 Matrix

Calculate the determinant of matrix $A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}$.

Solution:

$$\det(A) = (3 \times 4) - (1 \times 2) = 12 - 2 = 10$$

Answer: The determinant of matrix $A$ is 10.

Example 2: Determinant with Negative Elements

Calculate the determinant of matrix $B = \begin{bmatrix} -2 & 3 \\ 5 & -1 \end{bmatrix}$.

Solution:

$$\det(B) = ((-2) \times (-1)) - (3 \times 5) = 2 - 15 = -13$$

Answer: The determinant of matrix $B$ is -13.

Example 3: Zero Determinant

Calculate the determinant of matrix $C = \begin{bmatrix} 6 & 3 \\ 4 & 2 \end{bmatrix}$.

Solution:

$$\det(C) = (6 \times 2) - (3 \times 4) = 12 - 12 = 0$$

Answer: The determinant is 0. This matrix is singular (not invertible).

The determinant provides important information about a matrix. For $2 \times 2$ matrices (and square matrices in general):

• A non-zero determinant (like in Example 1 and 2) indicates that the matrix is invertible, meaning it has an inverse
• A zero determinant means the matrix is not invertible, or singular, and it does not have an inverse
• The absolute value of the determinant represents the scaling factor of area when the matrix is viewed as a linear transformation
• The sign of the determinant indicates whether the transformation preserves or reverses orientation

A $2 \times 2$ matrix $A$ is invertible (has an inverse) if and only if its determinant $\det(A) \neq 0$. If $\det(A) \neq 0$, the inverse of $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, denoted as $A^{-1}$, is given by:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

If $\det(A) = 0$, then matrix $A$ is singular and has no inverse.

Steps to Find the Inverse

To find the inverse of a $2 \times 2$ matrix:

1. Calculate the determinant $\det(A) = ad - bc$
2. If $\det(A) = 0$, the inverse does not exist
3. If $\det(A) \neq 0$, swap the positions of $a$ and $d$, change the signs of $b$ and $c$, and multiply the resulting matrix by $\frac{1}{\det(A)}$

Example 4: Finding the Inverse of a 2×2 Matrix

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

Step 1: Calculate the determinant.

From Example 1, $\det(A) = 10$. Since $\det(A) \neq 0$, the inverse exists.

Step 2: Apply the inverse formula.

$$A^{-1} = \frac{1}{10} \begin{bmatrix} 4 & -1 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix}$$

Answer: $A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix}$

Example 5: Matrix with No Inverse

Determine if matrix $S = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$ has an inverse.

Step 1: Calculate the determinant.

$$\det(S) = (2 \times 2) - (4 \times 1) = 4 - 4 = 0$$

Since $\det(S) = 0$, matrix $S$ is singular and does not have an inverse.

Example 6: Another Inverse Calculation

Find the inverse of $D = \begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix}$.

Solution:

$\det(D) = (5)(1) - (2)(3) = 5 - 6 = -1$

Since $\det(D) \neq 0$, the inverse exists:

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

To verify that $A^{-1}$ is indeed the inverse of $A$, we should check if $A \times A^{-1} = I$ and $A^{-1} \times A = I$, where $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is the $2 \times 2$ Identity Matrix.

Example 7: Verifying the Inverse

Let's verify if $A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix}$ is the inverse of $A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}$.

Check: $A \times A^{-1}$

After multiplication (detailed calculation omitted for brevity):

$$A \times A^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$$ ✓

The verification confirms that $A^{-1}$ is correct!

1️⃣ Solving Systems of Linear Equations

Matrix inverses can solve systems like:

$3x + y = 7$

$2x + 4y = 10$

Write as $AX = B$, then $X = A^{-1}B$

2️⃣ Computer Graphics Transformations

Inverse matrices undo transformations: rotations, scaling, shearing. Essential for animation and game development.

3️⃣ Cryptography

Hill cipher uses matrix inverses to encode and decode messages.

4️⃣ Economics & Optimization

Input-output models in economics use matrix inverses to predict economic changes.

5️⃣ Engineering & Physics

Solving circuit equations, stress analysis, and mechanical systems.

Example 8: Solving a System Using Inverse

Solve the system: $3x + y = 7$ and $2x + 4y = 10$

Solution:

Write as $\begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 10 \end{bmatrix}$

From Example 4, $A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix}$

$\begin{bmatrix} x \\ y \end{bmatrix} = A^{-1}\begin{bmatrix} 7 \\ 10 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

Answer: $x = 2, y = 1$