Matrices and Matrix Operations
1. Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.
4. Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.
For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
[latex]B=\left[\begin{array}{cc}2& 14\\ 22& 6\end{array}\right],C=\left[\begin{array}{cc}1& 5\\ 8& 92\\ 12& 6\end{array}\right],D=\left[\begin{array}{cc}10& 14\\ 7& 2\\ 5& 61\end{array}\right],E=\left[\begin{array}{cc}6& 12\\ 14& 5\end{array}\right][/latex]
7. [latex]C+D[/latex]
9. [latex]B-E[/latex]
11. [latex]D-B[/latex]
For the following exercises, use the matrices below to perform scalar multiplication.
[latex]B=\left[\begin{array}{rr}\hfill 3& \hfill 9\\ \hfill 21& \hfill 12\\ \hfill 0& \hfill 64\end{array}\right],C=\left[\begin{array}{rrrr}\hfill 16& \hfill 3& \hfill 7& \hfill 18\\ \hfill 90& \hfill 5& \hfill 3& \hfill 29\end{array}\right],D=\left[\begin{array}{rrr}\hfill 18& \hfill 12& \hfill 13\\ \hfill 8& \hfill 14& \hfill 6\\ \hfill 7& \hfill 4& \hfill 21\end{array}\right][/latex]
13. [latex]3B[/latex]
15. [latex]-4C[/latex]
17. [latex]100D[/latex]
For the following exercises, use the matrices below to perform matrix multiplication.
[latex]B=\left[\begin{array}{rrr}\hfill 3& \hfill 6& \hfill 4\\ \hfill -8& \hfill 0& \hfill 12\end{array}\right],C=\left[\begin{array}{rr}\hfill 4& \hfill 10\\ \hfill -2& \hfill 6\\ \hfill 5& \hfill 9\end{array}\right],D=\left[\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 12\\ \hfill 9& \hfill 3& \hfill 1\\ \hfill 0& \hfill 8& \hfill -10\end{array}\right][/latex]
19. [latex]BC[/latex]
21. [latex]BD[/latex]
23. [latex]CB[/latex]
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
[latex]A=\left[\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill 6& \hfill 7\end{array}\right], D=\left[\begin{array}{rrr}\hfill -8& \hfill 7& \hfill -5\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill 9& \hfill 2\end{array}\right],E=\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill 3\\ \hfill 7& \hfill -6& \hfill -5\\ \hfill 1& \hfill 0& \hfill 9\end{array}\right][/latex]
25. [latex]4A+5D[/latex]
27. [latex]3D+4E[/latex]
29. [latex]100D - 10E[/latex]
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])
[latex]A=\left[\begin{array}{rr}\hfill -10& \hfill 20\\ \hfill 5& \hfill 25\end{array}\right],B=\left[\begin{array}{rr}\hfill 40& \hfill 10\\ \hfill -20& \hfill 30\end{array}\right],C=\left[\begin{array}{rr}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right][/latex]
31. [latex]BA[/latex]
33. [latex]BC[/latex]
35. [latex]{B}^{2}[/latex]
37. [latex]{B}^{2}{A}^{2}[/latex]
39. [latex]{\left(AB\right)}^{2}[/latex]
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])
[latex]A=\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill -2& \hfill 3& \hfill 4\\ \hfill -1& \hfill 1& \hfill -5\end{array}\right],C=\left[\begin{array}{rr}\hfill 0.5& \hfill 0.1\\ \hfill 1& \hfill 0.2\\ \hfill -0.5& \hfill 0.3\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill -6& \hfill 7& \hfill 5\\ \hfill 4& \hfill 2& \hfill 1\end{array}\right][/latex]
41. [latex]AB[/latex]
43. [latex]BD[/latex]
45. [latex]{D}^{2}[/latex]
49. [latex]A\left(BC\right)[/latex]
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.
[latex]A=\left[\begin{array}{rrr}\hfill -2& \hfill 0& \hfill 9\\ \hfill 1& \hfill 8& \hfill -3\\ \hfill 0.5& \hfill 4& \hfill 5\end{array}\right],B=\left[\begin{array}{rrr}\hfill 0.5& \hfill 3& \hfill 0\\ \hfill -4& \hfill 1& \hfill 6\\ \hfill 8& \hfill 7& \hfill 2\end{array}\right],C=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right][/latex]
51. [latex]BA[/latex]
53. [latex]BC[/latex]
Solving Systems with Gaussian Elimination
1. Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.
For the following exercises, write the augmented matrix for the linear system.
7. [latex]\begin{array}{l}\text{ }16y=4\hfill \\ 9x-y=2\hfill \end{array}[/latex]
9. [latex]\begin{array}{l}\hfill \\ \text{ }x+5y+8z=19\hfill \\ \text{ }12x+3y=4\hfill \\ 3x+4y+9z=-7\hfill \end{array}[/latex]
For the following exercises, write the linear system from the augmented matrix.
11. [latex]\left[\left.\begin{array}{rr}\hfill -2& \hfill 5\\ \hfill 6& \hfill -18\end{array}\right\rvert\begin{array}{r}\hfill 5\\ \hfill 26\end{array}\right][/latex]
13. [latex]\left[\left.\begin{array}{rrr}\hfill 3& \hfill 2& \hfill 0\\ \hfill -1& \hfill -9& \hfill 4\\ \hfill 8& \hfill 5& \hfill 7\end{array}\right\rvert\begin{array}{r}\hfill 3\\ \hfill -1\\ \hfill 8\end{array}\right][/latex]
15. [latex]\left[\left.\begin{array}{rrr}\hfill 4& \hfill 5& \hfill -2\\ \hfill 0& \hfill 1& \hfill 58\\ \hfill 8& \hfill 7& \hfill -3\end{array}\right\rvert\begin{array}{r}\hfill 12\\ \hfill 2\\ \hfill -5\end{array}\right][/latex]
For the following exercises, solve the system by Gaussian elimination.
17. [latex]\left[\left.\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 1& \hfill 0\end{array}\right\rvert\begin{array}{r}\hfill 1\\ \hfill 2\end{array}\right][/latex]
19. [latex]\left[\left.\begin{array}{rr}\hfill -1& \hfill 2\\ \hfill 4& \hfill -5\end{array}\right\rvert\begin{array}{r}\hfill -3\\ \hfill 6\end{array}\right][/latex]
21. [latex]\begin{array}{l}\text{ }2x - 3y=-9\hfill \\ 5x+4y=58\hfill \end{array}[/latex]
25. [latex]\begin{array}{l}-5x+8y=3\hfill \\ 10x+6y=5\hfill \end{array}[/latex]
27. [latex]\begin{array}{l}-60x+45y=12\hfill \\ \text{ }20x - 15y=-4\hfill \end{array}[/latex]
31. [latex]\begin{gathered}\frac{3}{4}x-\frac{3}{5}y=4\\ \frac{1}{4}x+\frac{2}{3}y=1\end{gathered}[/latex]
33. [latex]\left[\left.\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right\rvert\begin{array}{r}\hfill 31\\ \hfill 45\\ \hfill 87\end{array}\right][/latex]
35. [latex]\left[\left.\begin{array}{rrr}\hfill 1& \hfill 2& \hfill 3\\ \hfill 0& \hfill 5& \hfill 6\\ \hfill 0& \hfill 0& \hfill 8\end{array}\right\rvert\begin{array}{r}\hfill 4\\ \hfill 7\\ \hfill 9\end{array}\right][/latex]
37. [latex]\begin{gathered}-2x+3y - 2z=3 \\ 4x+2y-z=9 \\ 4x - 8y+2z=-6\hfill \end{gathered}[/latex]
39. [latex]\begin{array}{l}\text{ }2x+3y+2z=1\hfill \\ \text{ }-4x - 6y - 4z=-2\hfill \\ \text{ }10x+15y+10z=5\hfill \end{array}[/latex]
43. [latex]\begin{array}{l}x+y+z=100\hfill \\ \text{ }x+2z=125\hfill \\ -y+2z=25\hfill \end{array}[/latex]
45. [latex]\begin{array}{l}-\frac{1}{2}x+\frac{1}{2}y+\frac{1}{7}z=-\frac{53}{14}\hfill \\ \text{ }\frac{1}{2}x-\frac{1}{2}y+\frac{1}{4}z=3\hfill \\ \text{ }\frac{1}{4}x+\frac{1}{5}y+\frac{1}{3}z=\frac{23}{15}\hfill \end{array}[/latex]
For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.
53. At a competing cupcake store, $4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost $2.25 and the red velvet cupcakes cost $1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day?
55. You invested $2,300 into account 1, and $2,700 into account 2. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.
57. A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for $86 each, with a delivery fee of $9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?
59. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.
61. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.
Solving Systems with Inverses
3. Explain whether a [latex]2\times 2[/latex] matrix with an entire row of zeros can have an inverse.
In the following exercises, show that matrix [latex]A[/latex] is the inverse of matrix [latex]B[/latex].
7. [latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right][/latex]
11. [latex]A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 0& 2\\ 1& 6& 9\end{array}\right],B=\frac{1}{4}\left[\begin{array}{ccc}6& 0& -2\\ 17& -3& -5\\ -12& 2& 4\end{array}\right][/latex]
For the following exercises, find the multiplicative inverse of each matrix, if it exists.
13. [latex]\left[\begin{array}{cc}3& -2\\ 1& 9\end{array}\right][/latex]
17. [latex]\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right][/latex]
19. [latex]\left[\begin{array}{cc}0.5& 1.5\\ 1& -0.5\end{array}\right][/latex]
21. [latex]\left[\begin{array}{ccc}0& 1& -3\\ 4& 1& 0\\ 1& 0& 5\end{array}\right][/latex]
25. [latex]\left[\begin{array}{ccc}\frac{1}{2}& \hfill\frac{1}{2}& \frac{1}{2}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}\\ \frac{1}{6}& \frac{1}{7}& \frac{1}{8}\end{array}\right][/latex]
For the following exercises, solve the system using the inverse of a [latex]2\times 2[/latex] matrix.
27. [latex]\begin{array}{l}\text{ }5x - 6y=-61\hfill \\ 4x+3y=-2\hfill \end{array}[/latex]
31. [latex]\begin{array}{l}-3x - 4y=9\hfill \\ 12x+4y=-6\hfill \end{array}[/latex]
For the following exercises, solve a system using the inverse of a [latex]3\text{}\times \text{}3[/latex]
matrix.
37. [latex]\begin{array}{l}\text{ }6x - 5y-z=31\hfill \\ \text{ }-x+2y+z=-6\hfill \\ \text{ }3x+3y+2z=13\hfill \end{array}[/latex]
41. [latex]\begin{array}{l}\frac{1}{2}x-\frac{1}{5}y+\frac{1}{5}z=\frac{31}{100}\hfill \\ -\frac{3}{4}x-\frac{1}{4}y+\frac{1}{2}z=\frac{7}{40}\hfill \\ -\frac{4}{5}x-\frac{1}{2}y+\frac{3}{2}z=\frac{1}{4}\hfill \end{array}[/latex]
For the following exercises, use a calculator to solve the system of equations with matrix inverses.
43. [latex]\begin{array}{l}2x-y=-3\hfill \\ -x+2y=2.3\hfill \end{array}[/latex]
45. [latex]\begin{array}{l}12.3x - 2y - 2.5z=2\hfill \\ 36.9x+7y - 7.5z=-7\hfill \\ 8y - 5z=-10\hfill \end{array}[/latex]
For the following exercises, find the inverse of the given matrix.
47. [latex]\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right][/latex]
49. [latex]\left[\begin{array}{rrrr}\hfill 1& \hfill -2& \hfill 3& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 2\\ \hfill 1& \hfill 4& \hfill -2& \hfill 3\\ \hfill -5& \hfill 0& \hfill 1& \hfill 1\end{array}\right][/latex]
For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.
55. Students were asked to bring their favorite fruit to class. 95% of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were 5% less popular than bananas, what are the percentages of each individual fruit?
57. A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?
59. Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?
61. Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. Jay picked 142 pieces of fruit weighing a total of 70 lb, 10 oz. He picked 15.5 times more oranges than pomegranates. How many of each fruit did Jay pick?
Solving Systems with Cramer’s Rule
1. Explain why we can always evaluate the determinant of a square matrix.
3. Explain what it means in terms of an inverse for a matrix to have a 0 determinant.
For the following exercises, find the determinant.
5. [latex]\left\rvert\begin{array}{cc}1& 2\\ 3& 4\end{array}\right\rvert[/latex]
7. [latex]\left\rvert\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill -1& \hfill 6\end{array}\right\rvert[/latex]
11. [latex]\left\rvert\begin{array}{cc}10& 0.2\\ 5& 0.1\end{array}\right\rvert[/latex]
15. [latex]\left\rvert\begin{array}{rrr}\hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -3\end{array}\right\rvert[/latex]
19. [latex]\left\rvert\begin{array}{rrr}\hfill -2& \hfill 1& \hfill 4\\ \hfill -4& \hfill 2& \hfill -8\\ \hfill 2& \hfill -8& \hfill -3\end{array}\right\rvert[/latex]
23. [latex]\left\rvert\begin{array}{rrr}\hfill 2& \hfill -1.6& \hfill 3.1\\ \hfill 1.1& \hfill 3& \hfill -8\\ \hfill -9.3& \hfill 0& \hfill 2\end{array}\right\rvert[/latex]
For the following exercises, solve the system of linear equations using Cramer’s Rule.
25. [latex]\begin{array}{l}2x - 3y=-1\\ 4x+5y=9\end{array}[/latex]
27. [latex]\begin{array}{l}\text{ }6x - 3y=2\hfill \\ -8x+9y=-1\hfill \end{array}[/latex]
33. [latex]\begin{array}{l}4x+10y=180\hfill \\ -3x - 5y=-105\hfill \end{array}[/latex]
For the following exercises, solve the system of linear equations using Cramer’s Rule.
35. [latex]\begin{array}{l}\text{ }x+2y - 4z=-1\hfill \\ \text{ }7x+3y+5z=26\hfill \\ -2x - 6y+7z=-6\hfill \end{array}[/latex]
37. [latex]\begin{array}{l}\text{ }4x+5y-z=-7\hfill \\ -2x - 9y+2z=8\hfill \\ \text{ }5y+7z=21\hfill \end{array}[/latex]
43. [latex]\begin{array}{l}\text{ }4x - 6y+8z=10\hfill \\ -2x+3y - 4z=-5\hfill \\ \text{ }x+y+z=1\hfill \end{array}[/latex]
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
53. You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?
55. A movie theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children’s tickets are $5.95, adult tickets are $11.15, and the total amount of revenue was $12,756, how many children’s tickets and adult tickets were sold?
57. You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was $29.50. If each gallon of yellow costs $2.59, and each gallon of blue costs $3.19, how many gallons of each color go into your green mix?
59. Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of tomato do you have?
61. At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.
63. A movie theatre sold tickets to three movies. The tickets to the first movie were $5, the tickets to the second movie were $11, and the third movie was $12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of $6,774. How many tickets for each movie were sold?