###### Preview Activity4.2.1

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?

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