91Ó°ÊÓ

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rrr} 0 & 1 & 1 \\ 1 & 2 & 3 \\ -1 & -1 & -2 \end{array}\right]$$

Let \(A\) and \(B\) be square matrices of order 3 such that \(|A|=10\) and \(|B|=12 .\) Find \((a)|A B|,(b)\left|A^{4}\right|,(c)|2 B|,(d)\left|(A B)^{T}\right|,\) and \((\mathrm{e})\left|A^{-1}\right|\)

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} 5 x_{1}-3 x_{2}+2 x_{3} &=2 \\ 2 x_{1}+2 x_{2}-3 x_{3} &=3 \\ x_{1}-7 x_{2}+8 x_{3} &=-4 \end{aligned}$$

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 area of the triangle having the given vertices. $$(-1,2),(2,2),(-2,4)$$

Find an equation of the line passing through the given points. $$(1,4),(3,4)$$

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. (a) In general, the determinant of the sum of two matrices equals the sum of the determinants of the matrices. (b) If \(A\) is a square matrix, then the determinant of \(A\) is equal to the determinant of the transpose of \(A\) (c) If the determinant of an \(n \times n\) matrix \(A\) is nonzero, then \(A \mathbf{x}=0\) has only the trivial solution.

The table below shows the projected values (in millions of dollars) of hardback college textbooks sold in the United States for the years 2007 to \(2009 .\) (Source: U.S. Census Bureau) $$\begin{array}{l|c} \hline \text {Year} & \text {Value} \\ \hline 2007 & 4380 \\ 2008 & 4439 \\ 2009 & 4524 \\ \hline \end{array}$$ (a) Create a system of linear equations for the data to fit the curve \(y=a t^{2}+b t+c,\) where \(t\) is the year and \(t=7\) corresponds to \(2007,\) and \(y\) is the value of the textbooks. (b) Use Cramer's Rule to solve your system. (c) Use a graphing utility to plot the data and graph your regression polynomial function. (d) Briefly describe how well the polynomial function fits the data.