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:
The product of a matrix and its inverse is the identity matrix, i.e., \(A A^{-1} = I\).
For a 2x2 matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), its inverse is given by \(A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\).
We can either use the property \(A A^{-1} = I\) or calculate the inverse of A and compare it with the given A\(^{-1}\).

Step 3: Detailed Explanation:

Method 1: Using the formula for the inverse
First, calculate the determinant of A:
\[ \det(A) = (x)(4) - (3)(2) = 4x - 6 \]
Now, find the inverse of A using the formula:
\[ A^{-1} = \frac{1}{4x-6} \begin{bmatrix} 4 & -3 \\ -2 & x \end{bmatrix} = \begin{bmatrix} \frac{4}{4x-6} & \frac{-3}{4x-6} \\ \frac{-2}{4x-6} & \frac{x}{4x-6} \end{bmatrix} \]
We are given:
\[ A^{-1} = \begin{bmatrix} -2 & 1.5 \\ 1 & -0.5 \end{bmatrix} \]
By comparing the corresponding elements of the calculated inverse and the given inverse, we can set up equations. Let's compare the element in the first row, first column:
\[ \frac{4}{4x-6} = -2 \]
\[ 4 = -2(4x - 6) \]
\[ 4 = -8x + 12 \]
\[ 8x = 12 - 4 \]
\[ 8x = 8 \]
\[ x = 1 \]

Method 2: Using \(A A^{-1} = I\)
\[ A A^{-1} = \begin{bmatrix} x & 3 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} -2 & 1.5 \\ 1 & -0.5 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
Let's compute the element in the first row, first column of the product:
\[ (x)(-2) + (3)(1) = 1 \]
\[ -2x + 3 = 1 \]
\[ -2x = 1 - 3 \]
\[ -2x = -2 \]
\[ x = 1 \]
Both methods yield the same result.

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