91Ó°ÊÓ

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{array}\right], \quad B=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $$

Use Cramer's rule to solve each system. $$ \begin{aligned}&2 x+2 y+3 z=10\\\&4 x-y+z=-5\\\&5 x-2 y+6 z=1\end{aligned} $$

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A(B+C) $$

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A-C $$

Determinants are used to show that three points lie on the same line (are collinear). If $$ \left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|=0 $$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal 0, then the points are not collinear. Use this information to work. Are the points \((-4,-6),(1,0),\) and \((11,12)\) collinear?