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The image you provided asks to show whether matrix \( B \) is the inverse of matrix \( A \) in two separate cases.

### Let's solve each part.

### Part (a):
Matrix \( A \) and \( B \) are given as:
\[
A = \begin{bmatrix} 2 & 4 \\ 2 & 0 \end{bmatrix}, \quad B = -\frac{1}{8} \begin{bmatrix} 0 & -4 \\ -2 & 2 \end{bmatrix}
\]

To check if \( B \) is the inverse of \( A \), we need to calculate \( A \times B \) and \( B \times A \). If both products equal the identity matrix \( I \), then \( B \) is indeed the inverse of \( A \).

### Part (b):
Matrix \( A \) and \( B \) are:
\[
A = \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

Similarly, to check if \( B \) is the inverse of \( A \), we calculate \( A \times B \) and \( B \times A \).

Let me calculate these and check for both parts.### Results:

#### Part (a):
For the matrices:
\[
A = \begin{bmatrix} 2 & 4 \\ 2 & 0 \end{bmatrix}, \quad B = -\frac{1}{8} \begin{bmatrix} 0 & -4 \\ -2 & 2 \end{bmatrix}
\]

Both \( A \times B \) and \( B \times A \) result in the identity matrix:
\[
A \times B = B \times A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]
This confirms that **B is the inverse of A** in part (a).

#### Part (b):
For the matrices:
\[
A = \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

The product \( A \times B \) is:
\[
A \times B = \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}
\]
And \( B \times A \) is also:
\[
B \times A = \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}
\]
Neither of these products results in the identity matrix, which means **B is not the inverse of A** in part (b).

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Would you like more details on the calculations, or have any further questions?

Here are 5 related questions for deeper understanding:
1. What is the formula to calculate the inverse of a 2x2 matrix?
2. How can we verify if two matrices are inverses of each other?
3. Why can't a matrix like \( A = \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix} \) have an inverse?
4. What happens if we multiply a matrix by its inverse from both sides?
5. Can every square matrix have an inverse?

**Tip**: A matrix has an inverse only if its determinant is non-zero.

Tunjukkan apakah matriks B merupakan invers A!

Explore how to determine if matrix B is the inverse of matrix A through detailed solutions for two distinct matrix pairs. This guide covers the fundamental concepts of matrix algebra, including matrix multiplication and inverse properties.

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