If $\left[ {\begin{array}{llllllllllllllllllll}2&1&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&0&{ - 1}\\{ - 1}&1&0\\0&1&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1\\0\\{ - 1}\end{array}} \right] = A$, then find the value of A.

We have, $\left[ {\begin{array}{llllllllllllllllllll}2&1&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&0&{ - 1}\\{ - 1}&1&0\\0&1&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1\\0\\{ - 1}\end{array}} \right] = A$

$\therefore$ $[2\,\,\,1\,\,\,\,3]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&0&{ - 1}\\{ - 1}&1&0\\0&1&1\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}{ - 2 - 1 + 0}&{0 + 1 + 3}&{ - 2 + 0 + 3}\end{array}} \right]$

$= [\begin{array}{llllllllllllllllllll}{ - 3}&4\end{array}\,\,\,\,1]$

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Now, $\left[ {\begin{array}{llllllllllllllllllll}{ - 3}&4&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1\\0\\{ - 1}\end{array}} \right] = A$

$\therefore$ $A = \left[ {\begin{array}{llllllllllllllllllll}{ - 3}&4&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1\\0\\{ - 1}\end{array}} \right]$

$= [ - 3 + 0 - 1] = [ - 4]$