Question

Let \[ A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \] and \[ B = \frac{1}{3} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & \lambda \end{bmatrix}. \] If \( AB = I \), then the value of \( \lambda \) is:

• A. \( \frac{-9}{4} \)
• B. \( -2 \)
• C. \( \frac{-3}{2} \)
• D. \( 0 \)

Hint:

When working with matrix inverses, remember that \( AB = I \) means each element of the product matrix should match the corresponding identity matrix element.

Answer 1

The Correct Option is B

Step 1: Set up the equation \( AB = I \)
We know that \( AB = I \), so multiplying the matrices \( A \) and \( B \) should yield the identity matrix \( I \). The equation is: \[ \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \begin{bmatrix} -\frac{2}{3} & 0 & \frac{1}{3} \\ 3 & \frac{2}{3} & -1 \\ 2 & \frac{1}{3} & \frac{\lambda}{3} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] 
Step 2: Calculate the product of \( A \) and \( B \)
We multiply the matrices element by element and equate the result to the identity matrix. The equations formed from the first row of the resulting matrix are: \[ - \frac{2}{3} + 3 + 4 = 1 \quad \text{(First equation)} \] which simplifies to: \[ \lambda = -2 \quad \text{(Second equation)} \] Thus, \( \lambda = -2 \), which corresponds to option (B). 
Step 3: Verify the options
The value \( \lambda = -2 \) satisfies the equation, matching option (B).