031 Review Part 2, Problem 8

Let  $A={\begin{bmatrix}1&3&8\\2&4&11\\1&2&5\end{bmatrix}}.$  Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}}   if possible.

Foundations:
To find the inverse of a matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,}   you augment the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
with the identity matrix and row reduce  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   to the identity matrix.

Solution:

Step 1:
We begin by augmenting the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   with the identity matrix. Hence, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 2 & 4 & 11 & 0 & 1 & 0\\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right].}

Step 2:
Now, we row reduce the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   to obtain the identity matrix. Hence, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 2 & 4 & 11 & 0 & 1 & 0\\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & -2 & -5 & -2 & 1 & 0\\ 0 & -1 & -3 & -1 & 0 & 1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & 1 & 3 & 1 & 0 & -1\\ 0 & -2 & -5 & -2 & 1 & 0 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & 1 & 3 & 1 & 0 & -1\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 0 & 1 & -8 & 8\\ 0 & 1 & 0 & 1 & -3 & 2\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & -2 & 1 & 2\\ 0 & 1 & 0 & 1 & -3 & 2\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right].} \end{array}}
Therefore, the inverse of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{ccc} -2 & 1 & 2\\ 1 & -3 & 2\\ 0 & 1 & -1 \end{array}\right]}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}=\left[\begin{array}{ccc} -2 & 1 & 2\\ 1 & -3 & 2\\ 0 & 1 & -1 \end{array}\right]}

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