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Chapter 1: Q13 (page 39)
Question: If A is a non-zero matrix of the form, then the rank of A must be 2.
Answer:
True, If A is a non-zero matrix of the form, then the rank of A is 2.
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Write the vector \(\left( {\begin{array}{*{20}{c}}5\\6\end{array}} \right)\) as the sum of two vectors, one on the line \(\left\{ {\left( {x,y} \right):y = {\bf{2}}x} \right\}\) and one on the line \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).
In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
32. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&{ - 3}&9&5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&0&0&{ - 1}\end{array}} \right]\)
In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).
11.\({a_1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\0\end{array}} \right],{a_2} = \left[ {\begin{array}{*{20}{c}}0\\1\\2\end{array}} \right],{a_3} = \left[ {\begin{array}{*{20}{c}}5\\{ - 6}\\8\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\\6\end{array}} \right]\)
Solve the systems in Exercises 11鈥�14.
12.\(\begin{aligned}{c}{x_1} - 3{x_2} + 4{x_3} = - 4\\3{x_1} - 7{x_2} + 7{x_3} = - 8\\ - 4{x_1} + 6{x_2} - {x_3} = 7\end{aligned}\)
Find the general solutions of the systems whose augmented matrices are given
11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).
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