Consider the matrices :
A=Consider the matrices :
`A=[(1,-2),(-1,3)]` and `B=[(a,b),(c,d)]`
If `AB=[(2,9),(5,6)]`, find the values of a,b,c and d.

Answer

To solve the problem, we need to find the values of \( a, b, c, \) and \( d \) in the matrix \( B \) given that the product \( AB \) equals a specific matrix.

Given:
\[ A = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad AB = \begin{pmatrix} 2 & 9 \\ 5 & 6 \end{pmatrix} \]

### Step 1: Calculate the product \( AB \)
To find \( AB \), we multiply the matrices \( A \) and \( B \):
\[ AB = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

Using matrix multiplication, we compute each element:
- First row, first column: \( 1 \cdot a + (-2) \cdot c = a - 2c \)
- First row, second column: \( 1 \cdot b + (-2) \cdot d = b - 2d \)
- Second row, first column: \( -1 \cdot a + 3 \cdot c = -a + 3c \)
- Second row, second column: \( -1 \cdot b + 3 \cdot d = -b + 3d \)

Thus, we have:
\[ AB = \begin{pmatrix} a - 2c & b - 2d \\ -a + 3c & -b + 3d \end{pmatrix} \]

### Step 2: Set up equations based on equality of matrices
Since \( AB \) is given to be equal to \( \begin{pmatrix} 2 & 9 \\ 5 & 6 \end{pmatrix} \), we can set up the following equations by comparing corresponding elements:
1. \( a - 2c = 2 \) (Equation 1)
2. \( b - 2d = 9 \) (Equation 2)
3. \( -a + 3c = 5 \) (Equation 3)
4. \( -b + 3d = 6 \) (Equation 4)

### Step 3: Solve the equations
From Equation 1:
\[ a = 2 + 2c \quad \text{(1)} \]

Substituting (1) into Equation 3:
\[ -(2 + 2c) + 3c = 5 \]
\[ -2 - 2c + 3c = 5 \]
\[ c = 7 \]

Now substitute \( c = 7 \) back into (1):
\[ a = 2 + 2(7) = 2 + 14 = 16 \]

Next, from Equation 2:
\[ b = 9 + 2d \quad \text{(2)} \]

Substituting (2) into Equation 4:
\[ -(9 + 2d) + 3d = 6 \]
\[ -9 - 2d + 3d = 6 \]
\[ d = 15 \]

Now substitute \( d = 15 \) back into (2):
\[ b = 9 + 2(15) = 9 + 30 = 39 \]

### Final Values
Thus, the values are:
\[ a = 16, \quad b = 39, \quad c = 7, \quad d = 15 \]

### Summary
The final values are:
- \( a = 16 \)
- \( b = 39 \)
- \( c = 7 \)
- \( d = 15 \)