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### Problem 4: Determinant and Inverse of Matrix \( A \)
We are given the matrix:

\[
A = \begin{pmatrix} 
3 & 0 & 1 \\ 
2 & -1 & 0 \\ 
4 & 5 & -2 
\end{pmatrix}
\]

#### Step 1: Compute the Determinant of Matrix \( A \)
To determine if matrix \( A \) is invertible, we first need to calculate its determinant \( \text{det}(A) \). The determinant for a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For matrix \( A \):

\[
\text{det}(A) = 3((-1)(-2) - 0 \times 5) - 0(2(-2) - 0 \times 4) + 1(2 \times 5 - (-1) \times 4)
\]

\[
= 3(2 - 0) - 0( -4 - 0) + 1(10 + 4)
\]

\[
= 3(2) + 1(14)
\]

\[
= 6 + 14 = 20
\]

Since \( \text{det}(A) = 20 \), which is not equal to zero, the matrix \( A \) is invertible.

#### Step 2: Find the Inverse of Matrix \( A \) using Cofactor Formula
The inverse of matrix \( A \), denoted by \( A^{-1} \), can be found using the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

Where \( \text{adj}(A) \) is the adjugate of \( A \), which is the transpose of the cofactor matrix of \( A \).

1. **Cofactor matrix**: The cofactor \( C_{ij} \) of an element \( a_{ij} \) is calculated by finding the determinant of the \( 2 \times 2 \) minor obtained by removing the \( i \)-th row and \( j \)-th column.

2. **Transpose the cofactor matrix**: After calculating all cofactors, take the transpose to get the adjugate matrix.

Let's compute the cofactors:

\[
C_{11} = \text{det}\begin{pmatrix} -1 & 0 \\ 5 & -2 \end{pmatrix} = (-1)(-2) - (0)(5) = 2
\]
\[
C_{12} = -\text{det}\begin{pmatrix} 2 & 0 \\ 4 & -2 \end{pmatrix} = -((2)(-2) - (0)(4)) = -(-4) = 4
\]
\[
C_{13} = \text{det}\begin{pmatrix} 2 & -1 \\ 4 & 5 \end{pmatrix} = (2)(5) - (-1)(4) = 10 + 4 = 14
\]

\[
C_{21} = -\text{det}\begin{pmatrix} 0 & 1 \\ 5 & -2 \end{pmatrix} = -((0)(-2) - (1)(5)) = 5
\]
\[
C_{22} =

Using the determinant criterion show that the following matrix is invertible and find its inverse using the cofactor formula.

Explore how to determine if a matrix is invertible using the determinant criterion and how to find its inverse using the cofactor formula. This detailed step-by-step solution covers key concepts in linear algebra, including matrix inversion and cofactor expansion.

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