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Chapter 6: Problem 73
Describe when the multiplication of two matrices is not defined.
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Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{rr}3 & -2 \\\1 & 5\end{array}\right], \quad B=\left[\begin{array}{rr}0 & 0 \\\5 & -6\end{array}\right]$$
Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$\text { Area }-\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|$$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use determinants to find the area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\)
What is the multiplicative identity matrix?
Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrr}1 & 1 & -1 \\\\-3 & 2 & -1 \\\3 & -3 & 2\end{array}\right]$$
Explain how to evaluate a second-order determinant.
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