91Ó°ÊÓ

For Exercises \(51-54,\) let $$ \mathbf{A}=\left[\begin{array}{rr} -1 & 0 \\ 2 & 1 \end{array}\right] \quad \text { and } \quad \mathbf{B}=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] $$ Show that $$ (\mathbf{A}+\mathbf{B})(\mathbf{A}-\mathbf{B}) \neq \mathbf{A}^{2}-\mathbf{B}^{2} $$ where $$ \mathbf{A}^{2}=\mathbf{A} \mathbf{A} \quad \text { and } \quad \mathbf{B}^{2}=\mathbf{B B} $$

Four solutions of the equation \(y=a x^{3}+b x^{2}+c x+d\) are given. Use a system of four equations in four variables to find the constants \(a, b, c,\) and \(d\) and write the equation. $$(-2,59),(-1,13),(1,-1), \text { and }(2,-17)$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, or logarithmic. $$f(x)=3$$

State the conditions under which \(\mathbf{A}^{-1}\) exists. Then find a formula for \(\mathbf{A}^{-1}\). $$\mathbf{A}=\left[\begin{array}{ll}x & 0 \\\0 & y\end{array}\right]$$

Four solutions of the equation \(y=a x^{3}+b x^{2}+c x+d\) are given. Use a system of four equations in four variables to find the constants \(a, b, c,\) and \(d\) and write the equation. $$(-2,-39),(-1,-12),(1,-6), \text { and }(3,16)$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, or logarithmic. $$f(x)=2-x-x^{2}$$

Determine whether the statement is true or false. In 2014 , the United States spent \(\$ 452\) billion more on its military than China spent on its military. Together, China and the United States spent \(\$ 710\) billion. (Source: International Institute for Strategic Studies) Find the amount spent on the military in China and in the United States.

For Exercises \(51-54,\) let $$ \mathbf{A}=\left[\begin{array}{rr} -1 & 0 \\ 2 & 1 \end{array}\right] \quad \text { and } \quad \mathbf{B}=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] $$ Show that $$\ (\mathbf{A}+\mathbf{B})(\mathbf{A}+\mathbf{B}) \neq \mathbf{A}^{2}+2 \mathbf{A} \mathbf{B}+\mathbf{B}^{2} $$

Simplify. Write answers in the form \(a+b i,\) where a and \(b\) are real numbers. $$\frac{3+i}{4-3 i}$$

State the conditions under which \(\mathbf{A}^{-1}\) exists. Then find a formula for \(\mathbf{A}^{-1}\). $$\mathbf{A}=\left[\begin{array}{lll}0 & 0 & x \\\0 & y & 0 \\\z & 0 & 0\end{array}\right]$$