Work each problem. Let A=[10000−101−1]. Show that A3i3, and use this result to find the inverse of A.

We have the matrix [10000−101−1] We need to show that A3i3 First find A×A A×A=[10000−101−1]×[10000−101−1] A2=[1000−110−10] A3=A2×A[1000−110−10]×[10000−101−1]=[100010001]=I3 ∀−1=I3 ∀−1=A3 ∀−1=A2A A−1=A2 Result A−1=[1000−110−10]

Expert answers to "Work each problem. Let A=[10000−101−1]. Show that A3i3,..."

High School
Math Word Problem

Lennie Carroll

Answered question

2021-07-03

Work each problem. Let $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & - 1 \end{matrix}]$ . Show that $A^{3} i_{3}$ , and use this result to find the inverse of A.

Answer & Explanation

Jozlyn

Skilled2021-07-04Added 85 answers

We have the matrix
$[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & - 1 \end{matrix}]$
We need to show that $A^{3} i_{3}$
First find $A \times A$
$A \times A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & - 1 \end{matrix}] \times [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & - 1 \end{matrix}]$
$A^{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 1 \\ 0 & - 1 & 0 \end{matrix}]$
$A^{3} = A^{2} \times A [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 1 \\ 0 & - 1 & 0 \end{matrix}] \times [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = I_{3}$
$\forall^{- 1} = I_{3}$
$\forall^{- 1} = A^{3}$
$\forall^{- 1} = A^{2} A$
$A^{- 1} = A^{2}$
Result $A^{- 1} = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 1 \\ 0 & - 1 & 0 \end{matrix}]$

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