91Ó°ÊÓ

Find the determinants of the matrices \(A\). $$\left[\begin{array}{cccc} 2 & 3 & 0 & 2 \\ 4 & 3 & 2 & 1 \\ 6 & 0 & 0 & 3 \\ 7 & 0 & 0 & 4 \end{array}\right]$$

Find the determinants of the matrices \(A\). $$\left[\begin{array}{ccccc} 5 & 4 & 0 & 0 & 0 \\ 6 & 7 & 0 & 0 & 0 \\ 3 & 4 & 5 & 6 & 7 \\ 2 & 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 0 & 1 \end{array}\right]$$

If \(A\) is an \(n \times n\) matrix and \(k\) is an arbitrary constant what is the relationship between det \(A\) and \(\operatorname{det}(k A) ?\)

Consider a \(2 \times 2\) matrix $$A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$ with column vectors $$\vec{v}=\left[\begin{array}{l} a \\ c \end{array}\right] \quad \text { and } \quad \vec{w}=\left[\begin{array}{l} b \\ d \end{array}\right]$$ We define the linear transformation $$\left.T(\vec{x})=\left[\begin{array}{ll} \operatorname{det}[\vec{x} & \vec{w} \\ \operatorname{det}[\vec{v} & \vec{x} \end{array}\right]\right]$$ from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) a. Find the standard matrix \(B\) of \(T .\) (Write the entries of \(B\) in terms of the entries \(a, b, c, d\) of \(A .\) b. What is the relationship between the determinants of \(A\) and \(B ?\) c. Show that \(B A\) is a scalar multiple of \(I_{2}\). What about A \(B\) ? d. If \(A\) is noninvertible (but nonzero), what is the relationship between the image of \(A\) and the kernel of \(B ?\) What about the kernel of \(A\) and the image of \(B ?\) e. If \(A\) is invertible, what is the relationship between \(B\) and \(A^{-1} ?\)

Consider an invertible \(2 \times 2\) matrix \(A\) with integer entries. a. Show that if the entries of \(A^{-1}\) are integers, then \(\operatorname{det} A=1\) or det \(A=-1\) b. Show the converse: If det \(A=1\) or \(\operatorname{det} A=-1\) then the entries of \(A^{-1}\) are integers.

A square matrix is called a permutation matrix if each row and each column contains exactly one entry \(1,\) with all other entries being 0. Examples are \(I_{n},\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]\) and the matrices considered in Exercises 53 and 56 What are the possible values of the determinant of a permutation matrix?

Throughout this exercise, consider the Fibonacci sequence \(f_{0}, f_{1}, f_{2}, \ldots\) recursively defined by \(f_{0}=0\) \(f_{1}=1,\) and \(f_{n+2}=f_{n}+f_{n+1}\) for all \(n=0,1,2, \ldots\) a. Find the Fibonacci numbers \(f_{0}, f_{1}, \ldots, f_{8}\) b. Consider the matrix \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right] .\) Prove by induction (see Appendix B.1) that \(A^{n}=\left[\begin{array}{cc}f_{n+1} & f_{n} \\ f_{n} & f_{n-1}\end{array}\right]\) for all \(n=1,2, \ldots\) c. Show that \(f_{n+1} f_{n-1}-f_{n}^{2}=(-1)^{n} .\) This equation is known as Cassini 's identity; it was discovered by the Italian/French mathematician and astronomer Giovanni Domenico Cassini, \(1625-1712\)