Problem 4
Matrices \(A\) and \(B\) are given. Compute \((A B)^{-1}\) and \(B^{-1} A^{-1}\). $$ A=\left[\begin{array}{ll} 2 & 4 \\ 2 & 5 \end{array}\right], \quad B=\left[\begin{array}{ll} 2 & 2 \\ 6 & 5 \end{array}\right] $$
Problem 4
Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{cc} -3 & -6 \\ 4 & 0 \end{array}\right] \\ B=\left[\begin{array}{cc} 48 & -30 \\ 0 & -8 \end{array}\right] \end{array} $$
Problem 4
Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{l} 2 \\ 0 \end{array}\right], \vec{y}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] $$
Problem 4
Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \vec{u}=\left[\begin{array}{ll} 0.6 & 0.8 \end{array}\right] \quad \vec{v}=\left[\begin{array}{l} 0.6 \\ 0.8 \end{array}\right] $$
Problem 5
A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{cccc} 0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5 \end{array}\right], \\ \vec{b}=\left[\begin{array}{c} 0 \\ 0 \end{array}\right], \vec{u}=\left[\begin{array}{c} 11 \\ 4 \\ -12 \\ 0 \end{array}\right], \\ \vec{v}=\left[\begin{array}{c} 9 \\ -12 \\ 0 \\ 12 \end{array}\right] \end{array} $$
Problem 5
Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{cc} 1 & -1 \\ 7 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} -3 & 2 \\ 5 & 9 \end{array}\right] $$ $$ 3(A-B)+B $$
Problem 5
Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ \begin{array}{l} A=\left[\begin{array}{ll} -1 & -2 \\ -2 & -3 \end{array}\right] \\ B=\left[\begin{array}{lll} 13 & 4 & 7 \\ 22 & 5 & 12 \end{array}\right] \end{array} $$
Problem 5
Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \vec{u}=\left[\begin{array}{lll} 1 & 2 & -1 \end{array}\right] \vec{v}=\left[\begin{array}{c} 2 \\ 1 \\ -1 \end{array}\right] $$
Problem 6
A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{cccc} 0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5 \end{array}\right] \\ \vec{b}=\left[\begin{array}{c} 48 \\ 36 \end{array}\right], \vec{u}=\left[\begin{array}{c} -17 \\ -16 \\ 0 \\ 0 \end{array}\right] \\ \vec{v}=\left[\begin{array}{c} -8 \\ -28 \\ 0 \\ 12 \end{array}\right] \end{array} $$
Problem 6
Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ A=\left[\begin{array}{cc} -4 & 1 \\ -1 & -2 \end{array}\right] $$ $$ B=\left[\begin{array}{ccc} -2 & -10 & 19 \\ 13 & 2 & -2 \end{array}\right] $$
