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Chapter 2: Q22Q (page 93)
If the equation \(Hx = c\) is inconsistent for some c in \({\mathbb{R}^{\bf{n}}}\), what can you say about the equation \(Hx = {\bf{0}}\)? Why?
The equation \(Hx = 0\) has a nontrivial solution because H is not invertible.
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1-6: Solve the equation Ax=b by using the LU factorization given for A. In Exercises 1 and 2, also solve \(Ax = b\) by ordinary row reduction.
4. \(A = \left[ {\begin{array}{*{20}{c}}2&{ - 2}&4\\1&{ - 3}&1\\3&7&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}0\\{ - 5}\\7\end{array}} \right]\)
\(A = \left[ {\begin{array}{*{20}{c}}1&0&0\\{\frac{1}{2}}&1&0\\{\frac{3}{2}}&{ - 5}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}2&{ - 2}&4\\0&{ - 2}&{ - 1}\\0&0&{ - 6}\end{array}} \right]\)
Let \(A = LU\) be an \(LU\) factorization. Explain why A can be row reduced to U using only replacement operations. (This fact is the converse of wht was proved in the text.)
Exercises 23-26 display a matrix A and echelon form of A. Find a basis for Col A and a basis for Nul A.
\[A = \left[ {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{5}}&{\bf{9}}&{ - {\bf{2}}}\\{\bf{6}}&{\bf{5}}&{\bf{1}}&{{\bf{12}}}\\{\bf{3}}&{\bf{4}}&{\bf{8}}&{ - {\bf{3}}}\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{6}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{1}}&{\bf{5}}&{ - {\bf{6}}}\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\end{array}} \right]\]
In exercise 5 and 6, compute the product \(AB\) in two ways: (a) by the definition, where \(A{b_{\bf{1}}}\) and \(A{b_{\bf{2}}}\) are computed separately, and (b) by the row-column rule for computing \(AB\).
\(A = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{2}}}\\{ - {\bf{3}}}&{\bf{0}}\\{\bf{3}}&{\bf{5}}\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}\\{\bf{2}}&{ - {\bf{1}}}\end{aligned}} \right)\)
1. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{8}}&{\bf{6}}\\{\bf{5}}&{\bf{4}}\end{aligned}} \right)\).
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