Foundations/Comprehension 1. Find the inverse (A−1) of the following 3×3 matrix, using either the row-operation method or the adjugate method (6 points): A=(231113112)

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Algebra
Algebra I
Algebra foundations

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2021-09-07

Foundations/Comprehension
1. Find the inverse $(A^{- 1})$ of the following $3 \times 3$ matrix, using either the row-operation method or the adjugate method (6 points):

$A = (\begin{matrix} 2 \\ 3 \\ 1 \end{matrix} \begin{matrix} 1 \\ 1 \\ 3 \end{matrix} \begin{matrix} 1 \\ 1 \\ 2 \end{matrix})$

Answer & Explanation

Laaibah Pitt

Skilled2021-09-08Added 98 answers

Invers of the following matrix by elementary slow bane formations
$A = [\begin{matrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ 1 & 3 & 2 \end{matrix}]$
Set $A = I A$
$\to [\begin{matrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ 1 & 3 & 2 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ 1 & 3 & 2 \end{matrix}]$
is... $[\begin{matrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ 1 & 3 & 2 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ A
$(A = I A)$
$↓$
by row bane formations, we will change it in identity matrix)
$R_{1} \to \frac{R_{1}}{2}$
$[\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ 3 & 1 & 1 \\ 1 & 3 & 2 \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ A
$R_{2} \to R_{2} - 3 R_{3}$ ; $R_{3} \to R_{3} - R_{1}$
$[\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ 0 & (1 - 9) & (1 - 6) \\ 0 & (3 - \frac{1}{2}) & (2 - \frac{1}{2}) \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & 1 & - 3 \\ - \frac{1}{2} & 0 & 1 \end{matrix}]$ A
$[\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ 0 & - 8 & - 5 \\ 0 & \frac{5}{2} & \frac{3}{2} \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & 1 & - 3 \\ - \frac{1}{2} & 0 & 1 \end{matrix}]$ A
$R_{2} \to \frac{R {2}}{- 8}$
$[\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ 0 & 1 & \frac{5}{8} \\ 0 & \frac{5}{2} & \frac{3}{2} \end{matrix}] = [\begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{- 8} & \frac{3}{8} \\ - \frac{1}{2} & 0 & 1 \end{matrix}]$ A
$R_{3} \to R_{3} - \frac{5}{2} - R_{2}$

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