Find the inverse of the following matrices ` [{:(3,-1),(-4,2):}](ii)[{:(2,-6),(1,-2):}]`

Explanation

To find the inverse of the given matrices, we will follow the steps outlined in the video transcript. Let's denote the first matrix as \( A \) and the second matrix as \( B \).

### Part 1: Finding the Inverse of Matrix \( A \)

Given:
\[ A = \begin{pmatrix} 3 & -1 \\ -4 & 2 \end{pmatrix} \]

**Step 1: Calculate the Determinant of \( A \)**

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:
\[ \text{det}(A) = ad - bc \]
For matrix \( A \):
\[ \text{det}(A) = (3)(2) - (-1)(-4) = 6 - 4 = 2 \]

**Step 2: Interchange the Diagonal Elements**

Interchanging the diagonal elements of \( A \):
\[ A' = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix} \]

**Step 3: Change the Sign of the Off-Diagonal Elements**

Changing the signs of the off-diagonal elements:
\[ A'' = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \]

**Step 4: Divide by the Determinant**

Now, divide each element of \( A'' \) by the determinant:
\[ A^{-1} = \frac{1}{\text{det}(A)} A'' = \frac{1}{2} \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} = \begin{pmatrix} 1 & \frac{1}{2} \\ 2 & \frac{3}{2} \end{pmatrix} \]

### Part 2: Finding the Inverse of Matrix \( B \)

Given:
\[ B = \begin{pmatrix} 2 & -6 \\ 1 & -2 \end{pmatrix} \]

**Step 1: Calculate the Determinant of \( B \)**

For matrix \( B \):
\[ \text{det}(B) = (2)(-2) - (-6)(1) = -4 + 6 = 2 \]

**Step 2: Interchange the Diagonal Elements**

Interchanging the diagonal elements of \( B \):
\[ B' = \begin{pmatrix} -2 & -6 \\ 1 & 2 \end{pmatrix} \]

**Step 3: Change the Sign of the Off-Diagonal Elements**

Changing the signs of the off-diagonal elements:
\[ B'' = \begin{pmatrix} -2 & 6 \\ -1 & 2 \end{pmatrix} \]

**Step 4: Divide by the Determinant**

Now, divide each element of \( B'' \) by the determinant:
\[ B^{-1} = \frac{1}{\text{det}(B)} B'' = \frac{1}{2} \begin{pmatrix} -2 & 6 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ -\frac{1}{2} & 1 \end{pmatrix} \]

### Final Answers:

The inverses of the matrices are:
\[ A^{-1} = \begin{pmatrix} 1 & \frac{1}{2} \\ 2 & \frac{3}{2} \end{pmatrix} \]
\[ B^{-1} = \begin{pmatrix} -1 & 3 \\ -\frac{1}{2} & 1 \end{pmatrix} \]