https://file.aiquickdraw.com/mathbot/file/17270938618911wpug6k3.jpg

Let's tackle each question one by one based on the image.

### **Question 4: Matrix Multiplication**
We are given two matrices \( A \) and \( B \):

\[
A = \begin{pmatrix}
1 & -1 & 3 \\
-1 & 3 & -2 \\
-1 & 2 & 2
\end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix}
2 & -1 & 2 \\
-4 & 2 & 3 \\
1 & 0 & 0
\end{pmatrix}
\]

We are asked to find the product \( B \cdot A \).

The matrix multiplication rule states:

\[
(B \cdot A)_{ij} = \sum_{k=1}^{n} B_{ik} A_{kj}
\]

I will calculate \( B \cdot A \).

### **Question 5: Solving the Linear Equation**
We are given the equation:

\[
\left| \begin{pmatrix}
x - 1 & 3 \\
2 & -1
\end{pmatrix} \right| = 4
\]

We need to find the value of \( x \) that satisfies this determinant equation.

The determinant of a 2x2 matrix is:

\[
\text{det} = (x - 1)(-1) - (2 \times 3)
\]

We solve this equation.

### **Question 6: Finding the Inverse of Matrix \( B \)**
We are given the matrix:

\[
B = \begin{pmatrix}
0 & 2 \\
1 & 4
\end{pmatrix}
\]

The inverse of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is:

\[
B^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

I'll calculate the inverse of matrix \( B \).

### **Question 7: Finding Minor and Cofactor**
We are given the matrix \( B \):

\[
B = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
8 & 9 & 1
\end{pmatrix}
\]

We are asked to find \( M_{12} \) (the minor of element in row 1, column 2) and \( C_{22} \) (the cofactor of the element in row 2, column 2).

- The **minor** is the determinant of the submatrix obtained by removing the respective row and column.
- The **cofactor** is calculated by applying a sign (based on the position) to the minor.

I'll now solve each of these questions step-by-step.Here are the detailed answers for each question:

### **Question 4: Matrix Multiplication \( B \cdot A \)**

The product of the matrices \( B \cdot A \) is:

\[
B \cdot A = \begin{pmatrix}
1 & -1 & 12 \\
-9 & 16 & -10 \\
1 & -1 & 3
\end{pmatrix}
\]

### **Question 5: Solving for \( x \)**

The equation we solved was:

\[
\left| \begin{pmatrix}
x - 1 & 3 \\
2 & -1
\end{pmatrix} \right| = 4
\]

The solution is:
\[
x = -9
\]

### **Question 6: Inverse of Matrix \( B \)**

The matrix \( B = \begin{pmatrix} 0 & 2 \\ 1 & 4 \end{pmatrix} \) has an inverse:

\[
B^{-1} = \begin{pmatrix}
-2 & 1 \\
0.5 & 0
\end{pmatrix}
\]

### **Question 7: Minor \( M_{12} \) and Cofactor \( C_{22} \)**

For matrix \( B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 8 & 9 & 1 \end{pmatrix} \):

- The **minor** \( M_{12} \) is the determinant of the submatrix formed by removing the first row and second column:

\[
M_{12} = -44
\]

- The **cofactor** \( C_{22} \) is calculated as the determinant of the submatrix formed by removing the second row and second column, with the appropriate sign adjustment:

\[
C_{22} = -23
\]

---

Do you need more detailed steps or any clarifications?  
Here are some related questions you might find helpful:

1. How do we compute the determinant of a larger matrix like 3x3 or 4x4?
2. How can you verify if a matrix has an inverse before calculating it?
3. What is the geometric interpretation of matrix multiplication?
4. Why does the sign pattern appear in the cofactor expansion?
5. What applications of matrix inverses are common in real-world scenarios?

**Tip**: When multiplying matrices, always check if their dimensions are compatible. The number of columns in the first matrix must match the number of rows in the second matrix.

Diketahui matriks A = [1 -1 3; -1 3 -2; -1 2 2] dan B = [2 -1 2; -4 2 3; 1 0 0], tentukan hasil perkalian matriks B . A! Tentukan nilai x yang memenuhi |x-1 3; 2 -1| = 4! Tentukan invers dari matriks B = [0 2; 1 4]! Diketahui matriks B = [1 2 3; 4 5 6; 8 9 1], Jika M_ij dan C_ij berturut-turut adalah minor dan kofaktor pada baris ke-i dan kolom ke-j, tentukan nilai dari M_12 dan C_22!

This problem involves matrix multiplication, solving determinants, finding the inverse of a matrix, and calculating the minor and cofactor of a matrix. Key mathematical concepts include matrix operations, the determinant formula, and cofactor expansion. Suitable for students in Grades 10-12.

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