Problem 20
Show that det \(\left[\begin{array}{lll}a+x & b+x & c+x \\ b+x & c+x & a+x \\\ c+x & a+x & b+x\end{array}\right]\) \(=(a+b+c+3 x)\left[(a b+a c+b c)-\left(a^{2}+b^{2}+c^{2}\right)\right.\)
Problem 20
Let \(A\) be an invertible \(n \times n\) matrix. a. Show that the eigenvalues of \(A\) are nonzero. b. Show that the eigenvalues of \(A^{-1}\) are precisely the numbers \(1 / \lambda,\) where \(\lambda\) is an eigenvalue of \(A\). c. Show that \(c_{A^{-1}}(x)=\frac{(-x)^{n}}{\operatorname{det} A} c_{A}\left(\frac{1}{x}\right)\).
Problem 21
Suppose \(\lambda\) is an eigenvalue of a square matrix \(A\) with eigenvector \(\mathbf{x} \neq \mathbf{0}\) a. Show that \(\lambda^{2}\) is an eigenvalue of \(A^{2}\) (with the same \(\mathbf{x})\) b. Show that \(\lambda^{3}-2 \lambda+3\) is an eigenvalue of \(A^{3}-2 A+3 I\) c. Show that \(p(\lambda)\) is an eigenvalue of \(p(A)\) for any nonzero polynomial \(p(x)\).
Problem 22
Find a polynomial \(p(x)\) of degree 2 such that: $$ \text { a. } p(0)=2, p(1)=3, p(3)=8 $$ b. \(p(0)=5, p(1)=3, p(2)=5\)
Problem 22
Show that $$ \text { det }\left[\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_{1} \\ 0 & 0 & \cdots & a_{2} & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & a_{n-1} & \cdots & * & * \\ a_{n} & \+ & \cdots & \+ & + \end{array}\right]=(-1)^{k} a_{1} a_{2} \cdots a_{n} $$ where either \(n=2 k\) or \(n=2 k+1,\) and \(+\) -entries are artitrary.
Problem 23
An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{m}=0\) for some \(m \geq 1\) a. Show that every triangular matrix with zeros on the main diagonal is nilpotent. b. If \(A\) is nilpotent, show that \(\lambda=0\) is the only eigenvalue (even complex) of \(A\). c. Deduce that \(c_{A}(x)=x^{n},\) if \(A\) is \(n \times n\) and nilpotent.
Problem 23
By expanding along the first column. show that: $$ \text { det }\left[\begin{array}{ccccccc} 1 & 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1 & 1 \\ 1 & 0 & 0 & 0 & \cdots & 0 & 1 \end{array}\right]=1+(-1)^{n+1} $$ if the matrix is \(n \times n, n>2\)
Problem 23
Find a polynomial \(p(x)\) of degree 3 such that: $$ \text { a. } p(0)=p(1)=1, p(-1)=4, p(2)=-5 $$ b. \(p(0)=p(1)=1, p(-1)=2, p(-2)=-3\)
Problem 24
An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{m}=0\) for some \(m \geq 1\) a. Show that every triangular matrix with zeros on the main diagonal is nilpotent. b. If \(A\) is nilpotent, show that \(\lambda=0\) is the only eigenvalue (even complex) of \(A\). c. Deduce that \(c_{A}(x)=x^{n},\) if \(A\) is \(n \times n\) and nilpotent.
Problem 24
Form matrix \(B\) from a matrix \(A\) by writing the columns of \(A\) in reverse order. Express det \(B\) in terms of \(\operatorname{det} A\).
