Problem 12
If \(A\) and \(B\) are \(n \times n\) matrices, if \(A B=\) \(-B A,\) and if \(n\) is odd, show that either \(A\) or \(B\) has no inverse.
Problem 12
Give an example of two diagonalizable matrices \(A\) and \(B\) whose \(\operatorname{sum} A+B\) is not diagonalizable.
Problem 12
Consider the recurrence $$ x_{k+2}=a x_{k+1}+b x_{k}+c $$ where \(c\) may not be zero. a. If \(a+b \neq 1\) show that \(p\) can be found such that, if we set \(y_{k}=x_{k}+p\), then \(y_{k+2}=a y_{k+1}+b y_{k}\). [Hence, the scquence \(x_{k}\) can be found provided \(y_{k}\) can be found by the methods of this section (or otherwise).] b. Use (a) to solve \(x_{k+2}=x_{k+1}+6 x_{k}+5\) where \(x_{0}=1\) and \(x_{1}=1\).
Problem 12
If \(A\) has three columns with only the top two entries nonzero, show that \(\operatorname{det} A=0\).
Problem 13
Show that \(\operatorname{det} A B=\operatorname{det} B A\) holds for any two \(n \times n\) matrices \(A\) and \(B\).
Problem 13
If \(A\) is diagonalizable and 1 and -1 are the only eigenvalues, show that \(A^{-1}=A\).
Problem 13
a. Find \(\operatorname{det} A\) if \(A\) is \(3 \times 3\) and \(\operatorname{det}(2 A)=6\). b. Under what conditions is \(\operatorname{det}(-A)=\operatorname{det} A\) ?
Problem 14
If \(A^{k}=0\) for some \(k \geq 1,\) show that \(A\) is not invertible.
Problem 14
Evaluate by first adding all other rows to the first row. $$ \begin{array}{l} \text { a. det }\left[\begin{array}{ccc} x-1 & 2 & 3 \\ 2 & -3 & x-2 \\ -2 & x & -2 \end{array}\right] \\ \text { b. det }\left[\begin{array}{ccc} x-1 & -3 & 1 \\ 2 & -1 & x-1 \\ -3 & x+2 & -2 \end{array}\right] \end{array} $$
Problem 14
If \(A\) is diagonalizable and 0 and 1 are the only eigenvalues, show that \(A^{2}=A\).
