Preview Activity 4.2.1

Let's begin by reviewing some important ideas that we have seen previously.

  1. Suppose that \(A\) is a square matrix and that the nonzero vector \(\xvec\) is a solution to the homogeneous equation \(A\xvec = \zerovec\text{.}\) What can we conclude about the invertibility of \(A\text{?}\)

  2. How does the determinant \(\det A\) tell us if there is a nonzero solution to the homogeneous equation \(A\xvec = \zerovec\text{?}\)

  3. Suppose that

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

    Find the determinant \(\det A\text{.}\) What does this tell us about the solution space to the homogeneous equation \(A\xvec = \zerovec\text{?}\)

  4. FInd a basis for \(\nul(A)\text{.}\)

  5. What is the relationship between the rank of a matrix and the dimension of its null space?