Chapter 2

A matrix \(A\) and vector \(\vec{b}\) are given. (a) Solve the equation \(A \vec{x}=\vec{O}\) (b) Solve the equation \(A \vec{x}=\vec{b}\). In each of the above, be sure to write your answer in vector format. Also, when possible, give 2 particular solutions to each equation. $$ \begin{array}{l} A=\left[\begin{array}{ccccc} 3 & 0 & -2 & -4 & 5 \\ 2 & 3 & 2 & 0 & 2 \\ -5 & 0 & 4 & 0 & 5 \end{array}\right], \\ \vec{b}=\left[\begin{array}{c} -1 \\ -5 \\ 4 \end{array}\right] \end{array} $$

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{cc} -1 & 5 \\ 6 & 7 \end{array}\right] \\ B=\left[\begin{array}{cccc} 5 & -3 & -4 & -4 \\ -2 & -5 & -5 & -1 \end{array}\right] \end{array} $$

A matrix \(A\) and vector \(\vec{b}\) are given. (a) Solve the equation \(A \vec{x}=\vec{O}\) (b) Solve the equation \(A \vec{x}=\vec{b}\). In each of the above, be sure to write your answer in vector format. Also, when possible, give 2 particular solutions to each equation. $$ \begin{array}{l} A=\left[\begin{array}{ccccc} -1 & 3 & 1 & -3 & 4 \\ 3 & -3 & -1 & 1 & -4 \\ -2 & 3 & -2 & -3 & 1 \end{array}\right], \\ \vec{b}=\left[\begin{array}{c} 1 \\ 1 \\ -5 \end{array}\right] \end{array} $$

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -1 & 2 & 1 \\ -1 & 2 & -1 \\ 0 & 0 & -2 \end{array}\right] \\ B=\left[\begin{array}{ccc} 0 & 0 & -2 \\ 1 & 2 & -1 \\ 1 & 0 & 0 \end{array}\right] \end{array} $$

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. $$ A=\left[\begin{array}{cc} 2 & 4 \\ -1 & -2 \end{array}\right], \vec{b}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -1 & 2 & 1 \\ -1 & 2 & -1 \\ 0 & 0 & -2 \end{array}\right] \\ B=\left[\begin{array}{ccc} 0 & 0 & -2 \\ 1 & 2 & -1 \\ 1 & 0 & 0 \end{array}\right] \end{array} $$

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. $$ A=\left[\begin{array}{cc} 2 & 4 \\ -1 & -2 \end{array}\right], \vec{b}=\left[\begin{array}{c} -6 \\ 3 \end{array}\right] $$

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -4 & 3 & 3 \\ -5 & -1 & -5 \\ -5 & 0 & -1 \end{array}\right] \\ B=\left[\begin{array}{ccc} 0 & 5 & 0 \\ -5 & -4 & 3 \\ 5 & -4 & 3 \end{array}\right] \end{array} $$

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. $$ A=\left[\begin{array}{cc} 2 & -5 \\ -4 & -10 \end{array}\right], \vec{b}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] $$

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -4 & -1 & 3 \\ 2 & -3 & 5 \\ 1 & 5 & 3 \end{array}\right] \\ B=\left[\begin{array}{ccc} -2 & 4 & 3 \\ -1 & 1 & -1 \\ 4 & 0 & 2 \end{array}\right] \end{array} $$