Problem 8
Show that: a. det \(\left[\begin{array}{ccc}p+x & q+y & r+z \\ a+x & b+y & c+z \\ a+p & b+q & c+r\end{array}\right]=2\) det \(\left[\begin{array}{lll}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\) b. det \(\left[\begin{array}{ccc}2 a+p & 2 b+q & 2 c+r \\ 2 p+x & 2 q+y & 2 r+z \\ 2 x+a & 2 y+b & 2 z+c\end{array}\right]=9\) det \(\left[\begin{array}{lll}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\)
Problem 9
In each case either prove the statement or give an example showing that it is false: a. \(\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B\). b. If det \(A=0\), then \(A\) has two equal rows. c. If \(A\) is \(2 \times 2,\) then \(\operatorname{det}\left(A^{T}\right)=\) det \(A\). d. If \(R\) is the reduced row-echelon form of \(A\), then \(\operatorname{det} A=\) det \(R .\) c. If \(A\) is \(2 \times 2,\) then \(\operatorname{det}(7 A)=49\) det \(A\). f. \(\operatorname{det}\left(A^{T}\right)=-\operatorname{det} A\). g. \(\operatorname{det}(-A)=-\operatorname{det} A\) h. If \(\operatorname{det} A=\) det \(B\) where \(A\) and \(B\) are the same size, then \(A=B\).
Problem 9
The annual yield of wheat in a certain country has been found to equal the average of the yield in the previous two years. If the yields in 1990 and 1991 were 10 and 12 million tons respectively, find a formula for the yield \(k\) years after \(1990 .\) What is the long-term average yield?
Problem 9
a. If \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 2\end{array}\right]\) and \(B=\left[\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right]\) verify that \(A\) and \(B\) are diagonalizable, but \(A B\) is not. b. If \(D=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) find a diagonalizable matrix \(A\) such that \(D+A\) is not diagonalizable.
Problem 10
Explain what can be said about \(\operatorname{det} A\) if: a. \(A^{2}=A\) b. \(A^{2}=I\) c. \(A^{3}=A\) d. \(P A=P\) and \(P\) is invertible e. \(A^{2}=u A\) and \(A\) is \(n \times n\) f. \(A=-A^{T}\) and \(A\) is \(n \times\) \(n\) g. \(A^{2}+I=0\) and \(A\) is \(n \times n\)
Problem 10
Find the general solution to the recurrence \(x_{k+1}=r x_{k}+c\) where \(r\) and \(c\) are constants. [Hint: Consider the cases \(r=1\) and \(r \neq 1\) separately. If \(r \neq 1\) you will need the identity \(1+r+r^{2}+\cdots+r^{n-1}=\frac{1-r^{n}}{1-r}\) for \(n \geq 1 .]\)
Problem 10
If \(A\) is an \(n \times n\) matrix, show that \(A\) is diagonalizable if and only if \(A^{T}\) is diagonalizable.
Problem 11
Consider the length 3 recurrence \(x_{k+3}=a x_{k}+b x_{k+1}+c x_{k+2}\) a. If \(\mathbf{v}_{k}=\left[\begin{array}{c}x_{k} \\ x_{k+1} \\\ x_{k+2}\end{array}\right]\) and \(A=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c\end{array}\right]\) show that \(\mathbf{v}_{k+1}=\bar{A} \mathbf{v}_{k}\) b. If \(\lambda\) is any eigenvalue of \(A,\) show that \(\mathbf{x}=\left[\begin{array}{c}1 \\ \lambda \\\ \lambda^{2}\end{array}\right]\) is a \(\lambda\) -eigenvector. [Hint: Show directly that \(A \mathbf{x}=\lambda \mathbf{x}\).] c. Generalize (a) and (b) to a recurrence $$ x_{k+4}=a x_{k}+b x_{k+1}+c x_{k+2}+d x_{k+3} $$ of length 4 .
Problem 11
If \(A\) is diagonalizable, show that each of the following is also diagonalizable. a. \(A^{n}, n \geq 1\) b. \(k A, k\) any scalar. c. \(p(A), p(x)\) any polynomial (Theorem 3.3.1) d. \(U^{-1} A U\) for any invertible matrix \(U\). e. \(k I+A\) for any scalar \(k\).
Problem 11
If \(\operatorname{det} A=2\), det \(B=-1\), and \(\operatorname{det} C=\) 3, find: $$ \begin{array}{l} \text { a. } \operatorname{det}\left[\begin{array}{lll} A & X & Y \\ 0 & B & Z \\ 0 & 0 & C \end{array}\right] & \text { b. det }\left[\begin{array}{lll} A & 0 & 0 \\ X & B & 0 \\ Y & Z & C \end{array}\right] \\ \text { c. } & \text { det }\left[\begin{array}{lll} A & X & Y \\ 0 & B & 0 \\ 0 & Z & C \end{array}\right] & \text { d. det }\left[\begin{array}{ccc} A & X & 0 \\ 0 & B & 0 \\ Y & Z & C \end{array}\right] \end{array} $$
