Bryant Ortiz
10/01/2023 · Senior High School
Given \( A=\left[\begin{array}{cc}5 & 4 \\ -1 & 6\end{array}\right] \), find \( A^{-1} \). If \( A^{-1} \) does not exist, write "DNE" in each box.
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Tutor-Verified Answer
Step-by-step Solution
Find the inverse matrix of \( \begin{bmatrix}5 & 4 \\ -1 & 6\end{bmatrix} \).
Matrices by following steps:
- step0: Find the matrix inverse:
\(\left[\begin{array}{rr}{5}&{4}\\{-1}&{6}\end{array}\right]\)
- step1: Evaluate the determinant:
\(34\)
- step2: Using the formula:
\(\frac{1}{5\times 6-4\left(-1\right)}\times \left[\begin{array}{rr}{6}&{-4}\\{1}&{5}\end{array}\right]\)
- step3: Evaluate:
\(\frac{1}{34}\times \left[\begin{array}{rr}{6}&{-4}\\{1}&{5}\end{array}\right]\)
- step4: Multiply the terms:
\(\left[\begin{array}{rr}{\frac{1}{34}\times 6}&{\frac{1}{34}\left(-4\right)}\\{\frac{1}{34}\times 1}&{\frac{1}{34}\times 5}\end{array}\right]\)
- step5: Evaluate:
\(\left[\begin{array}{rr}{\frac{3}{17}}&{-\frac{2}{17}}\\{\frac{1}{34}}&{\frac{5}{34}}\end{array}\right]\)
The inverse of matrix \( A \) is:
\[ A^{-1} = \left[\begin{array}{rr}\frac{3}{17} & -\frac{2}{17} \\ \frac{1}{34} & \frac{5}{34}\end{array}\right] \]
Quick Answer
\[ A^{-1} = \left[\begin{array}{rr}\frac{3}{17} & -\frac{2}{17} \\ \frac{1}{34} & \frac{5}{34}\end{array}\right] \]
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