If `[[2, 1], [3, 2]]A[[-3, 2], [5, -3]]=[[1, 0], [0, 1]]`, then A=

Options

A. `[[1, 1], [1,0]]`
B. `[[1, 0], [1, 1]]`
C. `[[0, 1], [1, 1]]`
D. 0

Answer

To solve the equation \( [[2, 1], [3, 2]] A [[-3, 2], [5, -3]] = [[1, 0], [0, 1]] \), we will follow these steps:

### Step 1: Define the matrices
Let:
- \( B = [[2, 1], [3, 2]] \)
- \( C = [[-3, 2], [5, -3]] \)
- \( I = [[1, 0], [0, 1]] \) (identity matrix)

The equation can be rewritten as:
\[ B A C = I \]

### Step 2: Pre-multiply by \( B^{-1} \) and post-multiply by \( C^{-1} \)
To isolate \( A \), we will pre-multiply both sides by \( B^{-1} \) and post-multiply by \( C^{-1} \):
\[ B^{-1} (B A C) C^{-1} = B^{-1} I C^{-1} \]
This simplifies to:
\[ A = B^{-1} I C^{-1} = B^{-1} C^{-1} \]

### Step 3: Calculate \( B^{-1} \)
To find \( B^{-1} \), we first calculate the determinant of \( B \):
\[ \text{det}(B) = (2)(2) - (1)(3) = 4 - 3 = 1 \]
Since the determinant is non-zero, \( B \) is invertible. The formula for the inverse of a \( 2 \times 2 \) matrix \( [[a, b], [c, d]] \) is:
\[ \frac{1}{ad - bc} [[d, -b], [-c, a]] \]
Applying this to matrix \( B \):
\[ B^{-1} = \frac{1}{1} [[2, -1], [-3, 2]] = [[2, -1], [-3, 2]] \]

### Step 4: Calculate \( C^{-1} \)
Next, we calculate the determinant of \( C \):
\[ \text{det}(C) = (-3)(-3) - (2)(5) = 9 - 10 = -1 \]
Since the determinant is non-zero, \( C \) is also invertible. Using the inverse formula:
\[ C^{-1} = \frac{1}{-1} [[-3, -2], [5, -3]] = [[3, 2], [-5, 3]] \]

### Step 5: Calculate \( A \)
Now we can find \( A \) by multiplying \( B^{-1} \) and \( C^{-1} \):
\[ A = B^{-1} C^{-1} = [[2, -1], [-3, 2]] [[3, 2], [-5, 3]] \]

### Step 6: Perform the multiplication
Calculating the product:
\[ A_{11} = 2 \cdot 3 + (-1) \cdot (-5) = 6 + 5 = 11 \]
\[ A_{12} = 2 \cdot 2 + (-1) \cdot 3 = 4 - 3 = 1 \]
\[ A_{21} = -3 \cdot 3 + 2 \cdot (-5) = -9 - 10 = -19 \]
\[ A_{22} = -3 \cdot 2 + 2 \cdot 3 = -6 + 6 = 0 \]
Thus, we have:
\[ A = [[11, 1], [-19, 0]] \]

### Step 7: Check the options
The options provided were:
1. \( [[1, 1], [1, 0]] \)
2. \( [[1, 0], [1, 1]] \)
3. \( [[0, 1], [1, 1]] \)
4. Null matrix

None of the options match the calculated matrix \( A \).

### Conclusion
The value of \( A \) is:
\[ A = [[11, 1], [-19, 0]] \]