031 Review Part 2, Problem 8

Let  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= \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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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  $A$  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]}

Return to Review Problems