Blake Howell
02/07/2024 · Junior High School
Find the inverse of the given matrix, if it exists. Use the algorithm for finding \( A^{-1} \) by row reducing [A I]. \( A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 4 & 1 & -4 \\ 2 & 3 & -4\end{array}\right] \) Set up the matrix [A I]. [A I] \( =\square \) (Type an integer or simplified fraction for each matrix element.)
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The augmented matrix [A I] is:
\[
\begin{bmatrix}
1 & 0 & -2 & 1 & 0 & 0 \\
4 & 1 & -4 & 0 & 1 & 0 \\
2 & 3 & -4 & 0 & 0 & 1
\end{bmatrix}
\]
The inverse of matrix A is:
\[
A^{-1} = \begin{bmatrix}
\frac{1}{3} & \frac{1}{2} & -\frac{1}{6} \\
\frac{7}{3} & -1 & -\frac{4}{3} \\
-\frac{5}{6} & \frac{1}{4} & -\frac{1}{12}
\end{bmatrix}
\]
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