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Chapter 2: Q2.9-23E (page 93)
In Exercises 19-24, justify each answer or construction.
If possible, construct a \({\bf{3}} \times {\bf{4}}\) matrix A such that dim Nul \(A = {\bf{2}}\) and dim Col \(A = {\bf{2}}\).
\(\left[ {\begin{array}{*{20}{c}} \bullet &*&*&*\\0& \bullet &*&*\\0&0&0&0\end{array}} \right]\)
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Suppose the second column of Bis all zeros. What can you
say about the second column of AB?
If a matrix \(A\) is \({\bf{5}} \times {\bf{3}}\) and the product \(AB\)is \({\bf{5}} \times {\bf{7}}\), what is the size of \(B\)?
The inverse of \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}&{\bf{0}}\\C&I&{\bf{0}}\\A&B&I\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}&{\bf{0}}\\Z&I&{\bf{0}}\\X&Y&I\end{array}} \right]\). Find X, Y, and Z.
Suppose AB = AC, where Band Care \(n \times p\) matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible.
Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{5}}\\{ - {\bf{3}}}&{\bf{1}}\end{aligned}} \right)\) and \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{5}}}\\{\bf{3}}&k\end{aligned}} \right)\). What value(s) of \(k\), if any will make \(AB = BA\)?
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