System of Equations With Matrices: Get Stronger

Introduction to Matrices and Matrix Operations

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

[latex]A=\left[\begin{array}{cc}1& 3\\ 0& 7\end{array}\right],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],F=\left[\begin{array}{cc}0& 9\\ 78& 17\\ 15& 4\end{array}\right][/latex]

  1. [latex]C + D[/latex]
  2. [latex]B - E[/latex]
  3. [latex]D - B[/latex]

For the following exercises, use the matrices below to perform scalar multiplication.

[latex]A=\left[\begin{array}{cc}4& 6\\ 13& 12\end{array}\right],B=\left[\begin{array}{cc}3& 9\\ 21& 12\\ 0& 64\end{array}\right],C=\left[\begin{array}{cccc}16& 3& 7& 18\\ 90& 5& 3& 29\end{array}\right],D=\left[\begin{array}{ccc}18& 12& 13\\ 8& 14& 6\\ 7& 4& 21\end{array}\right][/latex]

  1. [latex]3B[/latex]
  2. [latex]-4C[/latex]
  3. [latex]100D[/latex]

For the following exercises, use the matrices below to perform matrix multiplication.

[latex]A=\left[\begin{array}{cc}-1& 5\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}3& 6& 4\\ -8& 0& 12\end{array}\right],C=\left[\begin{array}{cc}4& 10\\ -2& 6\\ 5& 9\end{array}\right],D=\left[\begin{array}{ccc}2& -3& 12\\ 9& 3& 1\\ 0& 8& -10\end{array}\right][/latex]

  1. [latex]BC[/latex]
  2. [latex]BD[/latex]
  3. [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}{cc}2& -5\\ 6& 7\end{array}\right],B=\left[\begin{array}{cc}-9& 6\\ -4& 2\end{array}\right],C=\left[\begin{array}{cc}0& 9\\ 7& 1\end{array}\right],D=\left[\begin{array}{ccc}-8& 7& -5\\ 4& 3& 2\\ 0& 9& 2\end{array}\right],E=\left[\begin{array}{ccc}4& 5& 3\\ 7& -6& -5\\ 1& 0& 9\end{array}\right][/latex]

  1. [latex]4A + 5D[/latex]
  2. [latex]3D + 4E[/latex]
  3. [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}{cc}-10& 20\\ 5& 25\end{array}\right],B=\left[\begin{array}{cc}40& 10\\ -20& 30\end{array}\right],C=\left[\begin{array}{cc}-1& 0\\ 0& -1\\ 1& 0\end{array}\right][/latex]

  1. [latex]BA[/latex]
  2. [latex]BC[/latex]
  3. [latex]B^2[/latex]
  4. [latex]B^2A^2[/latex]
  5. [latex](AB)^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.

[latex]A=\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right],B=\left[\begin{array}{ccc}-2& 3& 4\\ -1& 1& -5\end{array}\right],C=\left[\begin{array}{cc}0.5& 0.1\\ 1& 0.2\\ -0.5& 0.3\end{array}\right],D=\left[\begin{array}{ccc}1& 0& -1\\ -6& 7& 5\\ 4& 2& 1\end{array}\right][/latex]

  1. [latex]AB[/latex]
  2. [latex]BD[/latex]
  3. [latex]D^2[/latex]
  4. [latex]D^3[/latex]
  5. [latex]A(BC)[/latex]

Solving System of Equations using Matrices

For the following exercises, write the augmented matrix for the linear system.

  1. [latex]\begin{array}{l} 16y=4 \\ 9x-y=2 \end{array}[/latex]
  2. [latex]\begin{array}{l} x+5y+8z=19 \ 12x+3y=4 \\ 3x+4y+9z=-7 \\ 6x+12y+16z=4 \end{array}[/latex]

For the following exercises, write the linear system from the augmented matrix.

  1. [latex]\left[\begin{array}{cc|c}-2& 5& 5\\ 6& -18& 26\end{array}\right][/latex]
  2. [latex]\left[\begin{array}{ccc|c}3& 2& 0& 3\\ -1& -9& 4& -1\\ 8& 5& 7& 8\end{array}\right][/latex]
  3. [latex]\left[\begin{array}{ccc|c}4& 5& -2& 12\\ 0& 1& 58& 2\\ 8& 7& -3& -5\end{array}\right][/latex]

For the following exercises, solve the system by Gaussian elimination.

  1. [latex]\begin{array}{l} 2x-3y=-9 \\ 5x+4y=58 \end{array}[/latex]
  2. [latex]\begin{array}{l} -5x+8y=3 \\ 10x+6y=5 \end{array}[/latex]
  3. [latex]\begin{array}{l} 2x-y=2 \\ 3x+2y=17 \end{array}[/latex]
  4. [latex]\begin{array}{l} -2x+3y-2z=3 \\ 4x+2y-z=9 \\ 4x-8y+2z=-6 \end{array}[/latex]
  5. [latex]\begin{array}{l} x+2y-z=1 \\ -x-2y+2z=-2 \\ 3x+6y-3z=3 \end{array}[/latex]
  6. [latex]\begin{array}{l} -\dfrac{1}{2}x+\dfrac{1}{2}y+\dfrac{1}{2}z=-\dfrac{53}{14} \\ \dfrac{1}{2}x-\dfrac{1}{2}y+\dfrac{1}{4}z=3 \\ \dfrac{1}{4}x+\dfrac{1}{5}y+\dfrac{1}{3}z=\dfrac{23}{15} \end{array}[/latex]

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.

  1. At Bakari’s competing cupcake store, [latex]$4,520[/latex] worth of cupcakes are sold daily. The chocolate cupcakes cost [latex]$2.25[/latex] and the red velvet cupcakes cost [latex]$1.75[/latex]. If the total number of cupcakes sold per day is [latex]2,200[/latex], how many of each flavor are sold each day?
  2. You invested [latex]$2,300[/latex] into account 1, and [latex]$2,700[/latex] into account 2. If the total amount of interest after one year is [latex]$254[/latex], and account 2 has [latex]1.5[/latex] times the interest rate of account 1, what are the interest rates? Assume simple interest rates.
  3. A major appliance store has agreed to order vacuums from a startup founded by college engineering students. The store would be able to purchase the vacuums for [latex]$86[/latex] each, with a delivery fee of [latex]$9,200[/latex], regardless of how many vacuums are sold. If the store needs to start seeing a profit after [latex]230[/latex] units are sold, how much should they charge for the vacuums?
  4. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up [latex]12 \%[/latex] of total ice cream sales. This year, the same three ice creams made up [latex]16.9 \%[/latex] of ice cream sales. The rocky road sales doubled, the banana sales increased by [latex]50 \%[/latex], and the pumpkin sales increased by [latex]20 \%[/latex]. 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.
  5. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were [latex]900[/latex] nuts in the bag. [latex]30 \%[/latex] of the almonds, [latex]20 \%[/latex] of the cashews, and [latex]10 \%[/latex] of the pistachios were eaten, and now there are [latex]770[/latex] nuts left in the bag. Originally, there were [latex]100[/latex] more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

Solving Systems with Inverses

In the following exercises, show that matrix [latex]A[/latex] is the inverse of matrix [latex]B[/latex].

  1. [latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right], B=\left[\begin{array}{cc}-2& 1\\ \dfrac{3}{2}& -\dfrac{1}{2}\end{array}\right][/latex]
  2. [latex]A=\left[\begin{array}{cc}-2& \dfrac{1}{2}\\ 3& -1\end{array}\right], B=\left[\begin{array}{cc}-2& -1\\ -6& -4\end{array}\right][/latex]
  3. [latex]A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 0& 2\\ 1& 6& 9\end{array}\right], B=\dfrac{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.

  1. [latex]\left[\begin{array}{cc}3& -2\\ 1& 9\end{array}\right][/latex]
  2. [latex]\left[\begin{array}{cc}-3& 7\\ 9& 2\end{array}\right][/latex]
  3. [latex]\left[\begin{array}{ccc}0& 1& -3\\ 4& 1& 0\\ 1& 0& 5\end{array}\right][/latex]
  4. [latex]\left[\begin{array}{ccc}1& 9& -3\\ 2& 5& 6\\ 4& -2& 7\end{array}\right][/latex]
  5. [latex]\left[\begin{array}{ccc}\dfrac{1}{2}& \dfrac{1}{2}& \dfrac{1}{2}\\ \dfrac{1}{3}& \dfrac{1}{4}& \dfrac{1}{5}\\ \dfrac{1}{6}& \dfrac{1}{7}& \dfrac{1}{8}\end{array}\right][/latex]

For the following exercises, solve the system using the inverse of a [latex]2 \times 2[/latex] matrix.

  1. [latex]\begin{array}{l} 5x-6y=-61 \\ 4x+3y=-2 \end{array}[/latex]
  2. [latex]\begin{array}{l} 3x-2y=6 \\ -x+5y=-2 \end{array}[/latex]
  3. [latex]\begin{array}{l} -3x-4y=9 \\ 12x+4y=-6 \end{array}[/latex]

For the following exercises, solve a system using the inverse of a [latex]3 \times 3[/latex] matrix.

  1. [latex]\begin{array}{l} 3x-2y+5z=21 \\ 5x+4y=37 \\ x-2y-5z=5 \end{array}[/latex]
  2. [latex]\begin{array}{l} 6x-5y-z=31 \\ -x+2y+z=-6 \\ 3x+3y+2z=13 \end{array}[/latex]
  3. [latex]\begin{array}{l} 4x-2y+3z=-12 \\ 2x+2y-9z=33 \\ 6y-4z=1 \end{array}[/latex]

For the following exercises, find the inverse of the given matrix.

  1. [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]
  2. [latex]\left[\begin{array}{cccc}1& -2& 3& 0\\ 0& 1& 0& 2\\ 1& 4& -2& 3\\ -5& 0& 1& 1\end{array}\right][/latex]
  3. [latex]\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 1& 1& 1& 1& 1& 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.

  1. Students were asked to bring their favorite fruit to class. [latex]95 \%[/latex] of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were [latex]5 \%[/latex] less popular than bananas, what are the percentages of each individual fruit?
  2. 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 [latex]$13.99[/latex], the beanie at [latex]$7.99[/latex], and the cowboy hat at [latex]$14.49[/latex]. If [latex]100[/latex] hats were sold this past quarter, [latex]$1,119[/latex] was taken in by sales, and the amount of beanies sold was [latex]10[/latex] more than cowboy hats, how many of each should the clothing store order to replace those already sold?
  3. Three roommates shared a package of [latex]12[/latex] ice cream bars, but no one remembers who ate how many. If Micah ate twice as many ice cream bars as Joe, and Albert ate three less than Micah, how many ice cream bars did each roommate eat?
  4. Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs [latex]8[/latex] oz, a lemon [latex]5[/latex] oz, and a pomegranate [latex]11[/latex] oz. Jay picked [latex]142[/latex] pieces of fruit weighing a total of [latex]70[/latex] lb, [latex]10[/latex] oz. He picked [latex]15.5[/latex] times more oranges than pomegranates. How many of each fruit did Jay pick?