Chapter 9

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{aligned} x-y &=1 \\ 6 x+y+20 z &=14 \\ y+3 z &=1 \end{aligned}\right. $$

Low-resolution digital photographs use \(262,144\) pixels in a \(512 \times 512\) grid. If you enlarge a low-resolution digital photograph enough, describe what will happen.

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{aligned} v &-3 x+z=-3 \\ w+y &=-1 \\ x &+z=& 7 \\ v+w-x+4 y &=-8 \\ v+w+x+y+z &=8 \end{aligned}\right. $$

We have seen that determinants can be used to solve linear equations, give areas of triangles in rectangular coordinates, and determine equations of lines. Not impressed with these applications? Members of the group should research an application of determinants that they find intriguing. The group should then present a seminar to the class about this application.

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{array}{r} {w+x+y+z=4} \\ {w+3 x-2 y+2 z=7} \\ {2 w+2 x+y+z=3} \\ {w-x+2 y+3 z=5} \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I added matrices of the same order by adding corresponding elements.

The length of a rectangle is 14 feet more than the width. If the perimeter of the rectangle is 72 feet, what are its dimensions? (Section \(1.3,\) Example 6 )

Use a coding matrix \(A\) of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for each message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$ \begin{array}{lllllllllllll} {A} & {R} & {R} & {I} & {V} & {E} & {D} & {-} & {S} & {A} & {F} & {E} & {L} & {Y} \end{array} $$$$ \begin{array}{llllllllllllll} {1} & {18} & {18} & {9} & {22} & {5} & {4} & {0} & {19} & {1} & {6} & {5} & {12} & {25} \end{array} $$

Use a coding matrix \(A\) of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for each message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$ \begin{array}{llllllllll} {A} & {R} & {T} & {-E} & {N} & {R} & {I} & {C} & {H} & {E} & {S} \end{array} $$$$ \begin{array}{ccccccccccc} {1} & {18} & {20} & {0} & {5} & {14} & {18} & {9} & {3} & {8} & {5} & {19} \end{array} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with two matrices that can be added but not multiplied.

Find all zeros of \(f(x)=x^{3}-4 x^{2}+x+6\) (Section \(3.4, \text { Example } 3)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with two matrices that can be multiplied but not added.

Verify the identity: $$ \sin 2 x+1=(\sin x+\cos x)^{2} $$ (Section \(6.3, \text { Examples } 3 \text { and } 6)\)

Find two matrices \(A\) and \(B\) such that \(A B=B A\)

Exercises \(81-83\) will help you prepare for the material covered in the first section of the next chapter. Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) a. Set \(y=0\) and find the \(x\) -intercepts. b. Set \(x=0\) and find the \(y\) -intercepts.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used matrix multiplication to represent a system of linear equations.

Consider a square matrix such that each element that is not on the diagonal from upper left to lower right is zero. Experiment with such matrices (call each matrix \(A\) ) by finding \(A A .\) Then write a sentence or two describing a method for multiplying this kind of matrix by itself.

Exercises \(81-83\) will help you prepare for the material covered in the first section of the next chapter. Divide both sides of \(25 x^{2}+16 y^{2}=400\) by 400 and simplify.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made an encoding error by selecting the wrong square invertible matrix.

If \(A B=-B A,\) then \(A\) and \(B\) are said to be anticommutative. Are \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right]\) and \(B=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right]\) anticommutative?

Exercises \(81-83\) will help you prepare for the material covered in the first section of the Complete the square and write the circle's equation in standard form: $$x^{2}+y^{2}-2 x+4 y=4$$ Then give the center and radius of the circle and graph the equation.

The interesting and useful applications of matrix theory are nearly unlimited. Applications of matrices range from representing digital photographs to predicting long-range trends in the stock market. Members of the group should research an application of matrices that they find intriguing. The group should then present a seminar to the class about this application.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible.

Solve: \(\quad 3^{2 x-8}=27 .\) (Section \(4.4,\) Example 1 )

Find the solution set and then use a calculator to obtain a decimal approximation to two decimal places for the solution: $$ 7^{x-3}=5^{2 x+4} $$ (Section 4.4, Example 4 )

Solve and graph the solution set on a number line: $$ |2 x+3| \leq 13 $$ (Section 1.7, Example 8)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((A+B)^{-1}=A^{-1}+B^{-1},\) assuming \(A, B,\) and \(A+B\) are invertible.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \left[\begin{array}{rr} {1} & {-3} \\ {-1} & {3} \end{array}\right] \text { is an invertible matrix. } $$

Will help you prepare for the material covered in the next section. Multiply: $$ \left[\begin{array}{ll} {a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}} \end{array}\right]\left[\begin{array}{ll} {1} & {0} \\ {0} & {1} \end{array}\right] $$ After performing the multiplication, describe what happens to the elements in the first matrix.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give an example of a \(2 \times 2\) matrix that is own inverse.

Will help you prepare for the material covered in the next section. Use Gauss-Jordan elimination to solve the system: $$ \left\\{\begin{aligned} -x-y-z &=1 \\ 4 x+5 y &=0 \\ y-3 z &=0 \end{aligned}\right. $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } A=\left[\begin{array}{ll} {3} & {5} \\ {2} & {4} \end{array}\right], \text { find }\left(A^{-1}\right)^{-1} $$

Will help you prepare for the material covered in the next section. Multiply and write the linear system represented by the following matrix multiplication: $$ \left[\begin{array}{lll} {a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \\ {z} \end{array}\right]=\left[\begin{array}{l} {d_{1}} \\ {d_{2}} \\ {d_{3}} \end{array}\right] $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find values of \(a\) for which the following matrix is not invertible: $$ \left[\begin{array}{cc} {1} & {a+1} \\ {a-2} & {4} \end{array}\right] $$

Solve: \(\log _{2} x+\log _{2}(x+2)=3\) (Section \(4.4, \text { Example } 7)\)

Solve: \(\log (x+4)-\log (x-2)=\log x\) (Section \(4.4, \text { Example } 8)\)

Solve the system: $$ \left\\{\begin{array}{rr} {x^{2}-2 y^{2}=} & {-1} \\ {2 x^{2}-y^{2}=} & {1} \end{array}\right. $$

Solve triangle \(A B C\) with \(A=20^{\circ}, b=60, c=68 .\) Round lengths of sides to the nearest tenth and angle measures to the nearest degree. (Graph cannot copy)

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$ 2(-5)-(-3)(4) $$

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$ \frac{2(-5)-1(-4)}{5(-5)-6(-4)} $$

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$ 2(-30-(-3))-3(6-9)+(-1)(1-15) $$