91Ó°ÊÓ

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{rrrr} \sqrt{2} & 0 & 2 \sqrt{2} & 0 \\ -4 \sqrt{2} & \sqrt{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 3 & 1 \end{array}\right]$$

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{lllll} 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \end{array}\right]$$

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ a & b & c & d \end{array}\right]$$

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{array}\right]$$

(a) Prove that \(\operatorname{rank}(A B) \leq \operatorname{rank}(A) .\) [Hint: Review Exercise 30 in Section 3.1 or use transposes and Exercise \(59(a) \cdot]\) (b) Give an example in which rank(AB) \(<\operatorname{rank}(A)\).

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{ll} 4 & 2 \\ 3 & 4 \end{array}\right] \text { over } \mathbb{Z}_{5}$$

(a) Prove that if \(U\) is invertible, then \(\operatorname{rank}(U A)=\) \(\operatorname{rank}(A) .\left[\text {Hint}: A=U^{-1}(U A) .\right]\) (b) Prove that if \(V\) is invertible, then \(\operatorname{rank}(A V)=\) \(\operatorname{rank}(A)\).

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{lll} 2 & 1 & 0 \\ 1 & 1 & 2 \\ 0 & 2 & 1 \end{array}\right] \text { over } \mathbb{Z}_{3}$$

Prove that an \(m \times n\) matrix \(A\) has rank 1 if and only if \(A\) can be written as the outer product uv of a vector \(\mathbf{u}\) in \(\mathbb{R}^{m}\) and \(\mathbf{v}\) in \(\mathbb{R}^{n}\).

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{lll} 1 & 5 & 0 \\ 1 & 2 & 4 \\ 3 & 6 & 1 \end{array}\right] \text { over } \mathbb{Z}_{7}$$