91Ó°ÊÓ

Sketch the parallelogram spanned by the vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) on graph paper. Estimate the area of your parallelogram using your sketch. Finally, compute the determinant of the matrix \(\left[\mathbf{v}_{1}, \mathbf{v}_{2}\right]\) and compare with your estimate. \(\mathbf{v}_{1}=\left(\begin{array}{r}-2 \\ 5\end{array}\right), \mathbf{v}_{2}=\left(\begin{array}{l}4 \\ 3\end{array}\right)\)

Let \(A\) be an arbitrary \(n \times n\) matrix. (a) If the \(i\) th row of \(A\) is a scalar multiple of the \(j\) th row, prove that the determinant of \(A\) is zero. State and prove a similar statement about the columns of \(A\). (b) Without computing the determinant, explain why each of the following matrices has a zero determinant. $$ \begin{aligned} &\left(\begin{array}{rrr} 1 & 2 & 3 \\ -1 & 1 & 4 \\ 0 & 0 & 0 \end{array}\right) \quad\left(\begin{array}{rrr} -1 & 2 & 0 \\ 3 & 4 & 0 \\ 5 & 2 & 0 \end{array}\right) \\ &\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 5 & 1 & 2 \end{array}\right) \quad\left(\begin{array}{rrr} -1 & -2 & 3 \\ 1 & 2 & 1 \\ 2 & 4 & 1 \end{array}\right) \end{aligned} $$

Let \(A\) be an arbitrary \(n \times n\) matrix. (a) If row \(i\) is a linear combination of the preceding rows, prove that the determinant of \(A\) is zero. State and prove a similar statement about the columns of \(A\). (b) Without computing the determinant, explain why each of the following matrices has a zero determinant. $$ \begin{aligned} &\left(\begin{array}{rrr} 1 & 2 & 3 \\ -1 & 1 & 1 \\ 0 & 3 & 4 \end{array}\right) \quad\left(\begin{array}{rrr} 1 & 2 & 3 \\ 3 & 0 & 3 \\ -1 & 1 & 0 \end{array}\right) \\ &\left(\begin{array}{rrr} 1 & 1 & 0 \\ -1 & 1 & 1 \\ 1 & 3 & 1 \end{array}\right) \quad\left(\begin{array}{rrr} 1 & 1 & 5 \\ -1 & 1 & 1 \\ 1 & 0 & 2 \end{array}\right) \end{aligned} $$

Calculate the determinant of the given matrix. Determine if the matrix has a nontrivial nullspace, and if it does find a basis for the nullspace. Determine if the column vectors in the matrix are linearly independent. \(\left(\begin{array}{rrr}1 & 1 & 2 \\ -1 & 1 & 5 \\ 1 & 0 & -1\end{array}\right)\)