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Math  /  Algebra

QuestionFind A−1A^{-1} by forming [A∥][A \|] and then using row operations to obtain [∥B][\| B], where A−1=[B]A^{-1}=[B]. Check that AA−1=IA A^{-1}=I and A−1A=IA^{-1} A=I. A=[44−103−1−1−41]A=\left[\begin{array}{rrr} 4 & 4 & -1 \\ 0 & 3 & -1 \\ -1 & -4 & 1 \end{array}\right]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A−1=A^{-1}= □\square (Type an integer or a simplified fraction for each matrix element.) B. The matrix does not have an inverse.

Studdy Solution
The matrix B B is the inverse A−1 A^{-1} :
A−1=[13013−13−1−83−1−4−8]A^{-1} = \left[\begin{array}{ccc} \frac{1}{3} & 0 & \frac{1}{3} \\ -\frac{1}{3} & -1 & -\frac{8}{3} \\ -1 & -4 & -8 \end{array}\right]
Verify that AA−1=I A A^{-1} = I and A−1A=I A^{-1} A = I by performing matrix multiplication.
The inverse matrix A−1 A^{-1} is:
[13013−13−1−83−1−4−8]\boxed{\left[\begin{array}{ccc} \frac{1}{3} & 0 & \frac{1}{3} \\ -\frac{1}{3} & -1 & -\frac{8}{3} \\ -1 & -4 & -8 \end{array}\right]}

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