Question

If A = \(\begin{bmatrix} x & 3 \\ 2 & 4 \end{bmatrix}\) and A\(^{-1}\) = \(\begin{bmatrix} -2 & 1.5 \\ 1 & -0.5 \end{bmatrix}\), then the value of x is

• A. -2
• B. 1
• C. 1.5
• D. -0.5

Hint:

Using the property \(A A^{-1} = I\) is often faster than calculating the inverse from scratch. You only need to calculate one or two elements of the product matrix to form an equation and solve for the unknown, saving valuable time in an exam.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a 2x2 matrix A containing an unknown variable x, and its inverse matrix A^{-1}. We need to find the value of x.

Step 2: Key Formula or Approach (Alternate Method):
Instead of using the inverse formula with determinant, use the fundamental property A × A^{-1} = I and multiply the matrices to form a simple equation.

Step 3: Detailed Explanation:
We know A × A^{-1} = I. Given: A = [[x, 3], [2, 4]] and A^{-1} = [[-2, 1.5], [1, -0.5]]. Multiply first row of A with first column of A^{-1}: (Element at position 1,1 of product) = x(-2) + 3(1) = -2x + 3. This must equal the (1,1) element of identity matrix I, which is 1. So: -2x + 3 = 1 → -2x = 1 - 3 → -2x = -2 → x = 1. No need to compute determinant or use inverse formula.

Step 4: Final Answer:
The value of x is 1.