Find the inverse \(\begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}\)
A. \(\begin{vmatrix} 2 & -\frac{3}{2} \\ -3 & -\frac{5}{2} \end{vmatrix}\)
B. \(\begin{vmatrix} 2 & -\frac{3}{2} \\ -3 & \frac{5}{2} \end{vmatrix}\)
C. \(\begin{vmatrix} 2 & \frac{3}{2} \\ -3 & \frac{5}{2} \end{vmatrix}\)
D. \(\begin{vmatrix} 2 & \frac{3}{2} \\ -3 & \frac{5}{2} \end{vmatrix}\)
Correct Answer: B
Explanation
Let A = \(\begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}\)
Then |A| = \(\begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}\) = 20 - 18 = 2
Hence A-1 = \(\frac{1}{|A|}\begin{pmatrix} 4 & -3 \\ -6 & 5 \end{pmatrix}\)
= \(\frac{1}{2}\begin{pmatrix} 4 & -3 \\ -6 & 5 \end{pmatrix}\)
= \(\begin{pmatrix} 4 \times 1/2 & -3 \times 1/2 \\ -6 \times 1/2 & 5 \times 1/2 \end{pmatrix}\)
= \(\begin{pmatrix} 2 & -\frac{3}{2} \\ -3 & \frac{5}{2} \end{pmatrix}\)
