The Matrix Equation

  1. Use the definition of \(A\vec x\) to write the matrix equation as a vector equation.   solution

    2. \( \left[\begin{array}{rrrr} {1} & {-2} & {4} & {2} \\ {3} & {1} & {-5} & {3} \end{array}\right] \left[\begin{array}{r} {5}\\{1} \\ {3} \\{-3} \end{array}\right] =\left[\begin{array}{r} {9}\\ {-8} \end{array}\right] \)

  2. Suppose that \(\vec {u}\) and \(\vec v\) are solutions of the homogenous linear system \(A\vec x =\vec 0\). Prove that \(2\vec u - 3 \vec v \) is also a solution.   proof

  3. Solve the matrix equation \(A\vec{x}=\vec{b}\) and write the solution as a vector.   solution

    4. \( A=\left[\begin{array}{rrr}
    {1} & {2} & {4} \\
    {0} & {1} & {5} \\
    {-2} & {-4} & {-3}
    \end{array}\right], \quad \vec{b}=\left[\begin{array}{r}
    {-2} \\
    {2} \\
    {9}
    \end{array}\right]
    \)