Problem 15
If \(A\) is diagonalizable and \(\lambda \geq 0\) for each eigenvalue of \(A,\) show that \(A=B^{2}\) for some matrix \(B\)
Problem 15
a. Find \(b\) if det \(\left[\begin{array}{rrr}5 & -1 & x \\ 2 & 6 & y \\ -5 & 4 & z\end{array}\right]=a x+b y+c z\). b. Find \(c\) if det \(\left[\begin{array}{rrr}2 & x & -1 \\ 1 & y & 3 \\ -3 & z & 4\end{array}\right]=a x+b y+c z\).
Problem 16
Show that no \(3 \times 3\) matrix \(A\) exists such that \(A^{2}+I=0\). Find a \(2 \times 2\) matrix \(A\) with this property.
Problem 16
If \(P^{-1} A P\) and \(P^{-1} B P\) are both diagonal, show that \(A B=B A .\) [Hint: Diagonal matrices commute.]
Problem 16
Find the real numbers \(x\) and \(y\) such that det \(A=0\) if: a. \(A=\left[\begin{array}{lll}0 & x & y \\ y & 0 & x \\ x & y & 0\end{array}\right]\) $$ \text { b. } A=\left[\begin{array}{rrr} 1 & x & x \\ -x & -2 & x \\ -x & -x & -3 \end{array}\right] $$ $$ \begin{array}{l} \text { c. } A=\left[\begin{array}{rrrr} 1 & x & x^{2} & x^{3} \\ x & x^{2} & x^{3} & 1 \\ x^{2} & x^{3} & 1 & x \\ x^{3} & 1 & x & x^{2} \end{array}\right] \\ \text { d. } A=\left[\begin{array}{llll} x & y & 0 & 0 \\ 0 & x & y & 0 \\ 0 & 0 & x & y \\ y & 0 & 0 & x \end{array}\right] \end{array} $$
Problem 17
Show that det \(\left[\begin{array}{llll}0 & 1 & 1 & 1 \\ 1 & 0 & x & x \\ 1 & x & 0 & x \\ 1 & x & x & 0\end{array}\right]=-3 x^{2}\)
Problem 17
Show that \(\operatorname{det}\left(A+B^{T}\right)=\operatorname{det}\left(A^{T}+B\right)\) for any \(n \times n\) matrices \(A\) and \(B\).
Problem 18
Let \(A\) be any \(n \times n\) matrix and \(r \neq 0\) a real number. a. Show that the eigenvalues of \(r A\) are precisely the numbers \(r \lambda,\) where \(\lambda\) is an eigenvalue of \(A\). b. Show that \(c_{r A}(x)=r^{n} c_{A}\left(\frac{x}{r}\right)\).
Problem 18
Let \(A\) and \(B\) be invertible \(n \times n\) matrices. Show that \(\operatorname{det} A=\operatorname{det} B\) if and only if \(A=U B\) where \(U\) is a matrix with \(\operatorname{det} U=1\).
Problem 19
Exercise 3.1 .19 Given the polynomial \(p(x)=a+b x+c x^{2}+d x^{3}+x^{4}\), the matrix \(C=\left[\begin{array}{rrrr}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -a & -b & -c & -d\end{array}\right]\) is called the com- panion matrix of \(p(x)\). Show that det \((x l-C)=p(x)\).
