Question

If \[ A= \begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix} \] then \(A^{-1}=\)

• A. \(\frac{1}{5}\begin{bmatrix}4 & -3\\-1 & 2\end{bmatrix}\)
• B. \(\frac{1}{5}\begin{bmatrix}4 & 3\\1 & 2\end{bmatrix}\)
• C. \(\frac{1}{3}\begin{bmatrix}4 & -3\\-1 & 2\end{bmatrix}\)
• D. \(\begin{bmatrix}4 & -3\\-1 & 2\end{bmatrix}\)

Hint:

For: \[ \begin{bmatrix} a & b c & d \end{bmatrix} \] inverse is: \[ \frac{1}{ad-bc} \begin{bmatrix} d & -b -c & a \end{bmatrix} \] Interchange diagonal entries and change signs of off-diagonal entries.

Answer 1

The Correct Option is A

To find the inverse of a 2x2 matrix \(A\), we use the formula:

\(A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\)

where the matrix \(A\) is:

\(A= \begin{bmatrix} a & b\\ c & d \end{bmatrix} = \begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix}\)

1. First, determine the determinant of matrix \(A\), denoted as \(\text{det}(A)\):
   \(\text{det}(A) = (2)(4) - (3)(1) = 8 - 3 = 5\)
2. Using the formula for the inverse, plug in the values:
   \(A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3\\ -1 & 2 \end{bmatrix}\)

Therefore, the inverse of matrix \(A\) is:

\(\frac{1}{5}\begin{bmatrix}4 & -3\\-1 & 2\end{bmatrix}\)

This matches the correct answer option provided.