Problem 28
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ll} 3 & 0 \\ 0 & -1 \end{array}\right] \\ A=\left[\begin{array}{ll} 2 & 4 \\ 6 & 8 \end{array}\right] \end{array} $$
Problem 28
A matrix \(A\) and vector \(\vec{b}\) are given. Solve the equation \(A \vec{x}=\vec{b},\) write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane. $$ A=\left[\begin{array}{cc} 2 & -5 \\ -4 & -10 \end{array}\right], \vec{b}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$
Problem 29
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ll} 4 & 0 \\ 0 & -3 \end{array}\right] \\ A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 2 \end{array}\right] \end{array} $$
Problem 30
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right] \\ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] \end{array} $$
Problem 31
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{array}\right] \\ A=\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 5 \end{array}\right] \end{array} $$
Problem 32
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ll} d_{1} & 0 \\ 0 & d_{2} \end{array}\right] \\ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \end{array} $$
Problem 33
A diagonal matrix \(D\) and a matrix \(A\) are given. Find the products \(D A\) and \(A D,\) where possible. $$ \begin{array}{l} D=\left[\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}\right] \\ A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \end{array} $$
Problem 34
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} 4 \\ 9 \end{array}\right] $$
Problem 35
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{cc} -1 & 4 \\ 7 & 3 \end{array}\right], \quad \vec{x}=\left[\begin{array}{c} 2 \\ -1 \end{array}\right] $$
Problem 36
A matrix \(A\) and a vector \(\vec{b}\) are given. Solve the equation \(A \vec{x}=\vec{b}\) using Theorem \(8 .\) $$ \begin{array}{l} A=\left[\begin{array}{ccc} 1 & -6 & 0 \\ 0 & 1 & 0 \\ 2 & -8 & 1 \end{array}\right] \\ \vec{b}=\left[\begin{array}{c} -69 \\ 10 \\ -102 \end{array}\right] \end{array} $$
