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Chapter 2: Q16Q (page 93)
Suppose Aand Bare \(n \times n\), Bis invertible, and ABis invertible. Show that Ais invertible. (Hint: Let C=AB, and solve this equation for A.)
It is proved that Ais invertible.
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In exercise 11 and 12, the matrices are all \(n \times n\). Each part of the exercise is an implication of the form 鈥淚f 鈥渟tatement 1鈥� then 鈥渟tatement 2鈥�.鈥滿ark the implication as True if the truth of 鈥渟tatement 2鈥漚lways follows whenever 鈥渟tatement 1鈥� happens to be true. An implication is False if there is an instance in which 鈥渟tatement 2鈥� is false but 鈥渟tatement 1鈥� is true. Justify each answer.
a. If the equation \[A{\bf{x}} = {\bf{0}}\] has only the trivial solution, then \(A\) is row equivalent to the \(n \times n\) identity matrix.
b. If the columns of \(A\) span \({\mathbb{R}^n}\), then the columns are linearly independent.
c. If \(A\) is an \(n \times n\) matrix, then the equation \(A{\bf{x}} = {\bf{b}}\) has at least one solution for each \({\bf{b}}\) in \({\mathbb{R}^n}\).
d. If the equation \[A{\bf{x}} = {\bf{0}}\] has a non trivial solution, then \[A\] has fewer than \(n\) pivot positions.
e. If \({A^T}\) is not invertible, then \(A\) is not invertible.
Exercises 23-26 display a matrix A and echelon form of A. Find a basis for Col A and a basis for Nul A.
\[A = \left[ {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{5}}&{\bf{9}}&{ - {\bf{2}}}\\{\bf{6}}&{\bf{5}}&{\bf{1}}&{{\bf{12}}}\\{\bf{3}}&{\bf{4}}&{\bf{8}}&{ - {\bf{3}}}\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{6}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{1}}&{\bf{5}}&{ - {\bf{6}}}\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\end{array}} \right]\]
Unless otherwise specified, assume that all matrices in these exercises are \(n \times n\). Determine which of the matrices in Exercises 1-10 are invertible. Use a few calculations as possible. Justify your answer.
10. [M] \[\left[ {\begin{array}{*{20}{c}}5&3&1&7&9\\6&4&2&8&{ - 8}\\7&5&3&{10}&9\\9&6&4&{ - 9}&{ - 5}\\8&5&2&{11}&4\end{array}} \right]\]
Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?
Exercises 1-4 display sets in \({\mathbb{R}^2}\). Assume the sets include the bounding lines. In each case, give a specific reason why the set H is not a subspace of \({\mathbb{R}^2}\). (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
2.
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