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Part 1
Find the​ inverse, if it​ exists, for the given matrix.
left bracket Start 3 By 3 Matrix 1st Row 1st Column 2 2nd Column 0 3rd Column negative 3 2nd Row 1st Column 0 2nd Column negative 1 3rd Column 0 3rd Row 1st Column 3 2nd Column 3 3rd Column negative 5 EndMatrix right bracket 2	0	−3	0	−1	0	3	3	−5 
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The inverse matrix is enter your response here. ​(Type a​ matrix, using an integer or simplified fraction for each matrix element. Do not factor out a scalar​ multiple.)  Superscript   Subscript 

We are given the matrix:

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

### Steps to find the inverse:

1. **Find the determinant** of the matrix. If the determinant is 0, the matrix has no inverse.
2. If the determinant is non-zero, **find the matrix of minors**.
3. **Form the matrix of cofactors**.
4. **Transpose** the cofactor matrix to get the adjugate matrix.
5. **Divide** the adjugate matrix by the determinant to get the inverse.

#### Step 1: Determinant of the matrix

The determinant of a 3x3 matrix is calculated as:

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

Where the matrix is structured as:
\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
For our matrix \( A \):

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

Here, \( a = 2, b = 0, c = -3, d = 0, e = -1, f = 0, g = 3, h = 3, i = -5 \). Applying these values to the determinant formula:

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

Simplifying:

\[
\text{det}(A) = 2(5 - 0) - 0 + (-3)(0 + 3)
\]
\[
\text{det}(A) = 10 - 9 = 1
\]

Since the determinant is 1, the matrix **does have an inverse**.

#### Step 2: Find the matrix of minors

For each element in the matrix, we calculate the determinant of the 2x2 submatrix formed by deleting the row and column of that element. The matrix of minors for matrix \( A \) is:

\[
\text{Minors}(A) = \begin{pmatrix}
\begin{vmatrix} -1 & 0 \\ 3 & -5 \end{vmatrix} & \begin{vmatrix} 0 & 0 \\ 3 & -5 \end{vmatrix} & \begin{vmatrix} 0 & -1 \\ 3 & 3 \end{vmatrix} \\
\begin{vmatrix} 0 & -3 \\ 3 & -5 \end{vmatrix} & \begin{vmatrix} 2 & -3 \\ 3 & -5 \end{vmatrix} & \begin{vmatrix} 2 & 0 \\ 3 & 3 \end{vmatrix} \\
\begin{vmatrix} 0 & -3 \\ 0 & 0 \end{vmatrix} & \begin{vmatrix} 2 & -3 \\ 0 & 0 \end{vmatrix} & \begin{vmatrix} 2 & 0 \\ 0 & -1 \end{vmatrix}
\end{pmatrix}
\]

Calculating the 2x2 determinants:

\[
\text{Minors}(A) = \begin{pmatrix}
5 & 0 & -3 \\
9 & 1 & 6 \\
0 & 0 & -2
\end{pmatrix}
\]

#### Step 3: Matrix of cofactors

The matrix of cofactors is obtained by applying a checkerboard pattern of signs (starting with + in the top left corner) to the matrix of minors:

\[
\text{Cofactors}(A) = \begin{pmatrix}
5 & 0 & 3 \\
-9 & 1 & -6 \\
0 & 0 & -2
\end{pmatrix}
\]

#### Step 4: Transpose of the cofactor matrix (Adjugate matrix)

The adjugate matrix is the transpose of the cofactor matrix:

\[
\text{Adj}(A) = \begin{pmatrix}
5 & -9 & 0 \\
0 & 1 & 0 \\
3 & -6 & -2
\end{pmatrix}
\]

#### Step 5: Inverse matrix

Since the determinant is 1, the inverse matrix is simply the adjugate matrix divided by the determinant:

\[
A^{-1} = \frac{1}{\text{det}(A)} \text{Adj}(A) = \begin{pmatrix}
5 & -9 & 0 \\
0 & 1 & 0 \\
3 & -6 & -2
\end{pmatrix}
\]

Thus, the inverse of the given matrix is:

\[
A^{-1} = \begin{pmatrix}
5 & -9 & 0 \\
0 & 1 & 0 \\
3 & -6 & -2
\end{pmatrix}
\]

### Would you like more details or have any questions?

Here are 5 related questions to explore this concept:
1. How do you verify the correctness of an inverse matrix?
2. What is the geometrical interpretation of the inverse of a matrix?
3. How does the determinant affect whether a matrix is invertible?
4. Can a matrix with determinant 1 always have an integer-valued inverse?
5. How would you compute the inverse of a larger (4x4) matrix?

**Tip:** For larger matrices, using row reduction (Gauss-Jordan elimination) is often more efficient than the cofactor method to find the inverse.

Find the inverse, if it exists, for the given matrix. 
Matrix: 
2  0 -3
0 -1  0
3  3 -5

Learn how to find the inverse of a 3x3 matrix with a detailed step-by-step solution. This problem covers key concepts in linear algebra, including matrix inversion and determinant calculation. Suitable for students in Grades 10-12.

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