If A =`[{:((1)/(3),2),(0,2x-3):}] and B =[{:(3,6),(0,-1):}] and AB=I_(2)` then value of x is

Options

A. -1
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \( x \) such that the product of matrices \( A \) and \( B \) equals the identity matrix \( I_2 \).

Given:
\[ A = \begin{pmatrix} \frac{1}{3} & 2 \\ 0 & 2x - 3 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix} \]
and \( AB = I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).

### Step 1: Calculate the product \( AB \)
We will multiply matrices \( A \) and \( B \).

\[ AB = \begin{pmatrix} \frac{1}{3} & 2 \\ 0 & 2x - 3 \end{pmatrix} \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix} \]

Using the matrix multiplication formula, we compute each element of the resulting matrix:

1. First row, first column:
\[ \frac{1}{3} \cdot 3 + 2 \cdot 0 = 1 + 0 = 1 \]

2. First row, second column:
\[ \frac{1}{3} \cdot 6 + 2 \cdot (-1) = 2 - 2 = 0 \]

3. Second row, first column:
\[ 0 \cdot 3 + (2x - 3) \cdot 0 = 0 + 0 = 0 \]

4. Second row, second column:
\[ 0 \cdot 6 + (2x - 3) \cdot (-1) = 0 - (2x - 3) = -2x + 3 \]

Thus, we have:
\[ AB = \begin{pmatrix} 1 & 0 \\ 0 & -2x + 3 \end{pmatrix} \]

### Step 2: Set the product equal to the identity matrix
We know that:
\[ AB = I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

This gives us the equations:
1. From the first row, first column: \( 1 = 1 \) (true, no information).
2. From the first row, second column: \( 0 = 0 \) (true, no information).
3. From the second row, first column: \( 0 = 0 \) (true, no information).
4. From the second row, second column: \( -2x + 3 = 1 \).

### Step 3: Solve for \( x \)
Now we solve the equation:
\[ -2x + 3 = 1 \]
Subtract 3 from both sides:
\[ -2x = 1 - 3 \]
\[ -2x = -2 \]
Now, divide by -2:
\[ x = 1 \]

### Conclusion
The value of \( x \) is \( 1 \).