Activity2.2.2Matrix multiplication

  1. Find the matrix product

    \begin{equation*} \left[ \begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -1 \\ 2 \amp 4 \amp -3 \amp -2 \\ -1 \amp -2 \amp 6 \amp 1 \\ \end{array} \right] \left[ \begin{array}{r} 3 \\ 1 \\ -1 \\ 1 \\ \end{array} \right] \text{.} \end{equation*}
  2. Suppose that \(A\) is the matrix

    \begin{equation*} \left[ \begin{array}{rrr} 3 \amp -1 \amp 0 \\ 0 \amp -2 \amp 4 \\ 2 \amp 1 \amp 5 \\ 1 \amp 0 \amp 3 \\ \end{array} \right] \text{.} \end{equation*}

    If \(A\xvec\) is defined, what is the dimension of the vector \(\xvec\) and what is the dimension of \(A\xvec\text{?}\)

  3. A vector whoses entries are all zero is denoted by \(\zerovec\text{.}\) If \(A\) is a matrix, what is the product \(A\zerovec\text{?}\)

  4. Suppose that \(I = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\) is the identity matrix and \(\xvec=\threevec{x_1}{x_2}{x_3}\text{.}\) Find the product \(I\xvec\) and explain why \(I\) is called the identity matrix.

  5. Suppose we write the matrix \(A\) in terms of its columns as

    \begin{equation*} A = \left[ \begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \\ \end{array} \right] \text{.} \end{equation*}

    If the vector \(\evec_1 = \left[\begin{array}{r} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right]\text{,}\) what is the product \(A\evec_1\text{?}\)

  6. Suppose that

    \begin{equation*} A = \left[ \begin{array}{rrrr} 1 \amp 2 \\ -1 \amp 1 \\ \end{array} \right], \bvec = \left[ \begin{array}{r} 6 \\ 0 \end{array} \right] \text{.} \end{equation*}

    Is there a vector \(\xvec\) such that \(A\xvec = \bvec\text{?}\)

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