https://file.aiquickdraw.com/mathbot/file/1727072006419z50qbe3k.jpg

The image contains two questions about matrix operations:

### Question 3:
- **Problem**: Calculate the product of two matrices:
  \[
  \begin{pmatrix}
  2 & -3 \\
  0 & 5 \\
  -1 & 3
  \end{pmatrix}
  \times
  \begin{pmatrix}
  0 & 3 \\
  -4 & 1
  \end{pmatrix}
  \]
- **Solution**: I'll compute the matrix multiplication step by step.

### Question 4:
- **Problem**: Find the inverse of matrix \( A \), where:
  \[
  A = \begin{pmatrix}
  -4 & 2 \\
  -5 & 3
  \end{pmatrix}
  \]
- **Solution**: I'll compute the inverse using the formula for the inverse of a 2x2 matrix.

Let me solve both questions.### Solutions:

#### Question 3:
The result of the matrix multiplication:
\[
\begin{pmatrix}
2 & -3 \\
0 & 5 \\
-1 & 3
\end{pmatrix}
\times
\begin{pmatrix}
0 & 3 \\
-4 & 1
\end{pmatrix}
=
\begin{pmatrix}
12 & 3 \\
-20 & 5 \\
-12 & 0
\end{pmatrix}
\]

#### Question 4:
The inverse of matrix \( A \), where:
\[
A = \begin{pmatrix}
-4 & 2 \\
-5 & 3
\end{pmatrix}
\]
is:
\[
A^{-1} = \begin{pmatrix}
-1.5 & 1 \\
-2.5 & 2
\end{pmatrix}
\]

Would you like more details or have any questions about these calculations?

Here are 5 related questions you can explore:
1. How is matrix multiplication defined for arbitrary-sized matrices?
2. How can you check if a matrix is invertible?
3. What are the conditions for a matrix to have an inverse?
4. How would the product of two matrices change if one matrix is not invertible?
5. What role does the determinant play in calculating the inverse of a matrix?

**Tip**: For a 2x2 matrix, the inverse can be calculated if the determinant is non-zero, and it follows the formula:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]
for \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).

Hasil perkalian matriks (2, -3; 0, 5; -1, 3) dengan (0, 3; -4, 1) dan invers dari matriks (-4, 2; -5, 3)

This problem explores the multiplication of a 3x2 and 2x2 matrix and the inversion of a 2x2 matrix. Learn how to compute the product of two matrices and find the inverse of a matrix using formulas for matrix multiplication and matrix inversion. Suitable for students in Grades 10-12.

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