91Ó°ÊÓ

The Determinant of a Matrix Product In Exercises \(1-6,\) find \((a)|A|,\) (b) \(|B|,\) (c) \(A B,\) and \((d)|A B| .\) Then verify that \(|\boldsymbol{A}||\boldsymbol{B}|=|\boldsymbol{A B}|\). $$A=\left[\begin{array}{rrrr}2 & 0 & 1 & 1 \\\1 & -1 & 0 & 1 \\\2 & 3 & 1 & 0 \\\1 & 2 & 3 & 0\end{array}\right], \quad B=\left[\begin{array}{rrrr}1 & 0 & -1 & 1 \\\2 & 1 & 0 & 2 \\\1 & 1 & -1 & 0 \\\3 & 2 & 1 & 0\end{array}\right]$$

Use either elementary row or column operations, or cofactor expansion, to find the determinant by hand. Then use a software program or a graphing utility to verify your answer. $$\left|\begin{array}{rrr}1 & 0 & 2 \\\\-1 & 1 & 4 \\\2 & 0 & 3\end{array}\right|$$

Use either elementary row or column operations, or cofactor expansion, to find the determinant by hand. Then use a software program or a graphing utility to verify your answer. $$\left|\begin{array}{rrr}-1 & 3 & 2 \\\0 & 2 & 0 \\\1 & 1 & -1\end{array}\right|$$

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{array}{r}x_{1}-3 x_{2}=2 \\\2 x_{1}+x_{2}=1\end{array}$$

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}x_{1}+x_{2}-x_{3}=& 4 \\\2 x_{1}-x_{2}+x_{3}=& 6 \\\3 x_{1}-2 x_{2}+2 x_{3}=& 0\end{aligned}$$

Find the value(s) of \(k\) such that \(A\) is singular. $$A=\left[\begin{array}{rrr}1 & k & 2 \\\\-2 & 0 & -k \\\3 & 1 & -4\end{array}\right]$$

Find the determinant of the elementary matrix. (Assume \(k \neq 0\).) $$\left[\begin{array}{lll}1 & 0 & 0 \\\k & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$

Find the value(s) of \(k\) such that \(A\) is singular. $$A=\left[\begin{array}{lll}0 & k & 1 \\\k & 1 & k \\\1 & k & 0\end{array}\right]$$

Find the determinant of the elementary matrix. (Assume \(k \neq 0\).) $$\left[\begin{array}{lll}1 & 0 & 0 \\\0 & 1 & 0 \\\0 & k & 1\end{array}\right]$$

Find the volume of the tetrahedron with the given vertices. $$(-3,-3,-3),(3,-1,-3),(-3,-1,-3),(-2,3,2)$$