Show that B is the inverse of A.
A = 

1	2	

3	4
,  B = 

−2	1	

 
3
2
− 
1
2
AB =





= I


BA =





= I




To show that matrix \( B \) is the inverse of matrix \( A \), we need to demonstrate that both \( AB = I \) and \( BA = I \), where \( I \) is the identity matrix. Here's how we can verify it:

### Matrix \( A \) and \( B \):
\[
A = \begin{pmatrix} 
1 & 2 \\ 
3 & 4 
\end{pmatrix}
, \quad B = \begin{pmatrix} 
-2 & 1 \\ 
1.5 & -0.5 
\end{pmatrix}
\]

### Step 1: Calculate \( AB \)

\[
AB = \begin{pmatrix} 
1 & 2 \\ 
3 & 4 
\end{pmatrix}
\begin{pmatrix} 
-2 & 1 \\ 
1.5 & -0.5 
\end{pmatrix}
\]

Now, perform the matrix multiplication:
\[
AB = \begin{pmatrix} 
(1 \times -2 + 2 \times 1.5) & (1 \times 1 + 2 \times -0.5) \\ 
(3 \times -2 + 4 \times 1.5) & (3 \times 1 + 4 \times -0.5) 
\end{pmatrix}
= \begin{pmatrix} 
0 & 0 \\ 
0 & 1 
\end{pmatrix}
= \begin{pmatrix} 
1 & 0 \\ 
0 & 1 
\end{pmatrix}
\]

This shows that \( AB = I \).

### Step 2: Calculate \( BA \)

\[
BA = \begin{pmatrix} 
-2 & 1 \\ 
1.5 & -0.5 
\end{pmatrix}
\begin{pmatrix} 
1 & 2 \\ 
3 & 4 
\end{pmatrix}
\]

Now, perform the matrix multiplication:
\[
BA = \begin{pmatrix} 
(-2 \times 1 + 1 \times 3) & (-2 \times 2 + 1 \times 4) \\ 
(1.5 \times 1 + -0.5 \times 3) & (1.5 \times 2 + -0.5 \times 4) 
\end{pmatrix}
= \begin{pmatrix} 
1 & 0 \\ 
0 & 1 
\end{pmatrix}
\]

This shows that \( BA = I \).

### Conclusion:
Since both \( AB = I \) and \( BA = I \), we have demonstrated that \( B \) is indeed the inverse of \( A \).

Would you like a more detailed breakdown of any steps or explanations?

---

Here are 5 related questions to explore further:
1. How do you find the inverse of a 2x2 matrix using a formula?
2. What conditions must a matrix meet to have an inverse?
3. Can every square matrix have an inverse?
4. What is the determinant of matrix \( A \) and why is it important for inverses?
5. How does matrix inversion work for larger matrices (e.g., 3x3)?

**Tip:** For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the inverse exists if \( ad - bc \neq 0 \). The inverse formula is \( A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \).

Show that B is the inverse of A.
A = 
1  2
3  4
B = 
−2  1
1.5  −0.5

Learn how to demonstrate that matrix B is the inverse of matrix A by verifying both AB = I and BA = I. This solution explores matrix algebra concepts, including matrix multiplication and the determinant formula, suitable for high school students in Grades 10-12.

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