Free math worksheets

Let \(A=\begin{pmatrix}1&-2\\0&3\end{pmatrix}\) and \(B=\begin{pmatrix}4&1\\5&-1\end{pmatrix}\). Find \(A+B\).

Let \(M=\begin{pmatrix}1&-3\\2&0\end{pmatrix}\). Find \(-2M\).

Two electronics stores record sales of two tablet models, \(T_1\) and \(T_2\), during two consecutive weeks. Rows represent weeks 1 and 2, and columns represent \(T_1\) and \(T_2\): \(A=\begin{pmatrix}24&18\\30&12\end{pmatrix}\), \(B=\begin{pmatrix}16&22\\25&20\end{pmatrix}\). a) Find the matrix \(S\) of combined sales for both stores. b) Find \(D=A-B\), and interpret a negative entry in context. c) State the dimension requirement for adding or subtracting two matrices. d) What would a \(2\times2\) zero matrix represent as a store’s sales matrix?

A farm offers three produce-box types—Basic, Family, and Gourmet—through weekly and biweekly subscriptions. Rows represent the two delivery schedules, and columns represent the box types in the order listed. The initial subscriber matrix is \(A=\begin{pmatrix}45&60&30\\25&40&20\end{pmatrix}\). a) How many subscribers ordered a Gourmet box in total? b) A promotion increases every category by \(20\%\). Find the new matrix \(A_{\mathrm{new}}\). c) Additional summer subscribers are represented by \(N=\begin{pmatrix}10&15&5\\5&10&8\end{pmatrix}\). Find the summer subscriber matrix \(A_{\mathrm{total}}\).

A \(2\times5\) matrix \(A=(a_{ij})\) is defined by \(a_{ij}=2^i j\). Find all entries and write the matrix.

Construct the \(3\times3\) matrix \(M=(m_{ij})\) defined by \(m_{ij}=0.5\) when \(i=j\), and \(m_{ij}=0.25\) when \(i\ne j\).

A factory uses three raw materials, \(G_1, G_2, G_3\), to make four intermediate products, \(Z_1, Z_2, Z_3, Z_4\). In the \(3\times4\) matrix \(A\), the entry \(a_{ij}\) is the number of units of raw material \(G_i\) needed for one unit of product \(Z_j\). Construct \(A\) from these entries: \(a_{23}=8\), \(a_{11}=4\), \(a_{34}=7\), \(a_{12}=3\), \(a_{31}=2\), \(a_{22}=5\), \(a_{14}=6\), \(a_{33}=1\), \(a_{24}=2\). All other entries are \(0\).

Transportation costs, in dollars per ton, between four warehouses \(L_1, L_2, L_3, L_4\) are represented by \(K=\begin{pmatrix}0&10&15&20\\10&0&8&12\\15&8&0&5\\20&12&5&0\end{pmatrix}\). 1) A truck follows the route \(L_1\to L_2\to L_4\to L_3\to L_1\). Find the total transportation cost per ton for the trip. 2) Find each row sum and interpret it. Which warehouse is most central with respect to transportation cost? 3) A toll increase adds \(\$2.50\) per ton to every trip between different warehouses. Find the new matrix \(K_{\mathrm{new}}\). Does it remain symmetric?

Find \(u\), \(v\), and \(w\) so that \(K=\begin{pmatrix}10&2u&v-3\\6&5&8\\4&w&7\end{pmatrix}\) is symmetric about its main diagonal.

A company uses three raw materials, \(R_1, R_2, R_3\), to make two intermediate products, \(Z_1, Z_2\). Those intermediate products are then used to make two final products, \(E_1, E_2\). The raw-material requirements per unit of intermediate product are \(A=\begin{pmatrix}3&1\\2&4\\0&5\end{pmatrix}\), and the intermediate-product requirements per unit of final product are \(B=\begin{pmatrix}2&6\\3&1\end{pmatrix}\). Find \(C=AB\), which gives the direct raw-material requirements per unit of each final product.

A bakery makes three kinds of rolls. The ingredient amounts per roll are shown below: <table> <tr> <td></td> <th colspan="3">Roll type</th> </tr> <tr> <td></td> <td>Whole-grain</td> <td>Multigrain</td> <td>Pumpkin seed</td> </tr> <tr> <td>Flour (g)</td> <td>150</td> <td>120</td> <td>100</td> </tr> <tr> <td>Water (mL)</td> <td>90</td> <td>80</td> <td>70</td> </tr> <tr> <td>Seeds (g)</td> <td>10</td> <td>30</td> <td>25</td> </tr> </table> An order calls for \(60\) whole-grain rolls, \(50\) multigrain rolls, and \(40\) pumpkin seed rolls. Use matrix-vector multiplication to find the total amounts of flour, water, and seeds needed.

Let \(A=\begin{pmatrix}1&2\\0&3\end{pmatrix}\) and \(B=\begin{pmatrix}4&-1\\2&5\end{pmatrix}\). a) Find \(AB\) and \(BA\). b) Determine whether multiplication is commutative for these matrices. c) Find \(A^2\).

A catering company offers three menus: Veggie, Classic, and Premium. Ingredient amounts per serving are shown in pounds: <table> <tr><th>Menu</th><th>Rice (lb)</th><th>Vegetables (lb)</th><th>Protein (lb)</th></tr> <tr><td>Veggie</td><td>\(0.2\)</td><td>\(0.4\)</td><td>\(0\)</td></tr> <tr><td>Classic</td><td>\(0.15\)</td><td>\(0.2\)</td><td>\(0.15\)</td></tr> <tr><td>Premium</td><td>\(0.1\)</td><td>\(0.3\)</td><td>\(0.25\)</td></tr> </table> Ingredient prices per pound are represented by \(\mathbf{p}=\begin{pmatrix}2.00\\3.50\\14.00\end{pmatrix}\). Construct the ingredient matrix \(M\), and use matrix-vector multiplication to find the ingredient cost of one serving of each menu.

Use matrix properties to evaluate the vector expression efficiently: \(\begin{pmatrix}0.6&0.1\\0.4&0.9\end{pmatrix}\begin{pmatrix}14\\37\end{pmatrix}+\begin{pmatrix}0.6&0.1\\0.4&0.9\end{pmatrix}\begin{pmatrix}86\\63\end{pmatrix}\).

Evaluate the expression efficiently: \(\begin{pmatrix}0.8&0.4\\0.2&0.6\end{pmatrix}\begin{pmatrix}12\\18\end{pmatrix}-\begin{pmatrix}0.3&0.4\\0.2&0.1\end{pmatrix}\begin{pmatrix}12\\18\end{pmatrix}\).

Let \(A=\begin{pmatrix}3&-2\\4&1\end{pmatrix}\). a) Find a matrix \(X\) such that \(AX=A\). b) Verify by direct multiplication whether your matrix from part a also satisfies \(XA=A\). c) Use matrix dimensions to explain why the \(3\times3\) identity matrix \(I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\) cannot serve as an identity matrix for \(A\).

Matrices used for transition processes include several special matrix types. Investigate the additive identity matrix. 1. Find the entries of a \(2\times2\) matrix \(N=\begin{pmatrix}n_{11}&n_{12}\\n_{21}&n_{22}\end{pmatrix}\) such that \(M+N=M\) for every \(2\times2\) matrix \(M=\begin{pmatrix}m_{11}&m_{12}\\m_{21}&m_{22}\end{pmatrix}\). 2. Explain why an additive identity matrix exists for every matrix dimension \(m\times n\). 3. A square matrix is column-stochastic when all entries are between \(0\) and \(1\), inclusive, and every column sum is \(1\). Can the additive identity from part 1 be a column-stochastic transition matrix? Explain.

A population transition matrix is \(M=\begin{pmatrix}0.7&0.2\\0.3&0.8\end{pmatrix}\). a) Suppose every entry is doubled proportionally. Find \(M_{\mathrm{new}}=2M\). b) Verify the distributive property by comparing \(3(M+M)\) with \(3M+3M\). c) Explain why matrix addition is commutative, using the commutative property of real-number addition. d) Explain why \(M_{\mathrm{new}}\) is not valid as a column-stochastic transition matrix.

A sporting-goods company ships four running-shoe models—Air, Speed, Trail, and Comfort—to three distribution centers—West, South, and East. Rows represent the centers in that order, and columns represent the models in the order listed. April shipments are \(L_A=\begin{pmatrix}120&150&80&200\\100&120&110&180\\90&130&100&150\end{pmatrix}\). a) How many Trail shoes were shipped in total? Which center received the greatest total number of shoes? b) May, June, and July each have the same shipments as April. Find the matrix \(L_Q\) for the three-month total. c) August shipments are \(L_{\mathrm{Aug}}=\begin{pmatrix}140&160&90&210\\110&130&120&190\\100&140&110&160\end{pmatrix}\). Find the combined shipment matrix for July and August.

Let \(A=\begin{pmatrix}4&-1&2\\0&3&-5\\1&2&1\end{pmatrix}\), \(B=\begin{pmatrix}-2&5&0\\1&-4&2\\3&0&-1\end{pmatrix}\), and \(C=\begin{pmatrix}1&2&-3\\-2&0&4\\0&1&5\end{pmatrix}\). Find: a) \(A+B-C\) b) \(2A-3B\) c) \(B-2(C-A)\)

Let \(M=\begin{pmatrix}1.5&-2&4\\0&3.2&-1\end{pmatrix}\) and \(N=\begin{pmatrix}-0.5&4&0\\2&-1.6&3\end{pmatrix}\). Find: a) \(2M+4N\) b) \(M-0.5N\) c) the matrix \(X\) satisfying \(3N+X=M\).

A delivery company serves five locations, \(A, B, C, D, E\). The matrix \(D=(d_{ij})\) gives driving distances in miles from location \(i\) to location \(j\): \(D=\begin{pmatrix}\cdots&4.0&\cdots&7.0&\cdots\\\cdots&\cdots&3.0&\cdots&6.0\\5.0&\cdots&\cdots&8.0&\cdots\\\cdots&2.0&\cdots&\cdots&9.0\\1.0&\cdots&4.0&\cdots&\cdots\end{pmatrix}\). 1) Complete \(D\), assuming the distance is the same in both directions and the distance from a location to itself is \(0\). 2) State the mathematical properties that characterize this distance matrix. 3) Interpret a row vector in this context.

An exchange process among three states is represented by \(P=\begin{pmatrix}0.5&0.3&0.2\\x&0.4&y\\0.2&0.3&0.5\end{pmatrix}\). a) Find \(x\) and \(y\) so that \(P\) is symmetric. b) Using those values, determine whether \(P\) is row-stochastic, meaning that every row sum is \(1\). c) State the general component condition for a square matrix \(A\) to be symmetric.

A furniture manufacturer makes tables \(T\), chairs \(C\), and benches \(B\). The component requirements per item are \(M_{CP}=\begin{pmatrix}8&5&4\\24&16&12\\2&0&4\end{pmatrix}\), where rows represent wood panels, screws, and hinges, and columns represent \(T, C, B\). Two customer orders are shown below: <table> <tr> <td></td> <td>Customer 1</td> <td>Customer 2</td> </tr> <tr> <td>Tables (\(T\))</td> <td>10</td> <td>5</td> </tr> <tr> <td>Chairs (\(C\))</td> <td>40</td> <td>20</td> </tr> <tr> <td>Benches (\(B\))</td> <td>15</td> <td>10</td> </tr> </table> Find the component requirements for both orders and present the result as a matrix.

Let \(M=\begin{pmatrix}3&-1\\2&4\end{pmatrix}\), \(N=\begin{pmatrix}1&0&5\\-2&3&1\end{pmatrix}\), and \(P=\begin{pmatrix}2\\-3\\0\end{pmatrix}\). Find every defined product formed from two of these matrices, allowing the same matrix to be used twice.

Let \(A=\begin{pmatrix}5&-2&0\\1&4&7\\-3&8&2\\6&0&-1\end{pmatrix}\) and \(P=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}\). a) Find \(AP\). b) Describe how right-multiplication by \(P\) changes the columns of \(A\). c) Find a matrix \(Q\) such that \(QA\) swaps the first and fourth rows of \(A\) while leaving the second and third rows unchanged.

Two consecutive changes in a dynamic system are represented by \(M=\begin{pmatrix}0.8&0.4\\0.2&0.6\end{pmatrix}\) and \(N=\begin{pmatrix}0.5&0\\0.5&1\end{pmatrix}\). Determine whether the order of the changes affects the overall result by finding \(MN\) and \(NM\).

A catering company delivers two lunch boxes, Box A and Box B, to three clients, Client X, Client Y, and Client Z. The material cost and preparation time for each box are shown below. <table> <tr><td></td><td>Box A</td><td>Box B</td></tr> <tr><td>Material cost (dollars)</td><td>\(5\)</td><td>\(8\)</td></tr> <tr><td>Preparation time (minutes)</td><td>\(12\)</td><td>\(15\)</td></tr> </table> The client orders for the first two weeks of the month are shown below. Week 1, \(B_1\): <table> <tr><td></td><td>Client X</td><td>Client Y</td><td>Client Z</td></tr> <tr><td>Box A</td><td>\(40\)</td><td>\(25\)</td><td>\(30\)</td></tr> <tr><td>Box B</td><td>\(20\)</td><td>\(30\)</td><td>\(10\)</td></tr> </table> Week 2, \(B_2\): <table> <tr><td></td><td>Client X</td><td>Client Y</td><td>Client Z</td></tr> <tr><td>Box A</td><td>\(20\)</td><td>\(35\)</td><td>\(10\)</td></tr> <tr><td>Box B</td><td>\(15\)</td><td>\(20\)</td><td>\(25\)</td></tr> </table> Find the total material cost and total preparation time for each client over both weeks in two different ways. State the matrix property that connects the methods.

A three-stage process is described by the transition matrices \(M_1=\begin{pmatrix}0.7&0.4\\0.3&0.6\end{pmatrix}\) from stage 1 to stage 2 and \(M_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}\) from stage 2 to stage 3. a) Find the matrix \(M_{\text{total}}\) for the direct transition from stage 1 to stage 3. b) Interpret the identity transition matrix \(M_2\) in the context of movement between two states. c) For an arbitrary matrix \(M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), prove by multiplication that the identity matrix \(I=\begin{pmatrix}1&0\\0&1\end{pmatrix}\) commutes with \(M\).

A publisher sells two professional magazines, \(Z_1\) and \(Z_2\), in two regions, \(R_1\) and \(R_2\). The matrices \(G_1\) and \(G_2\) give the profit in dollars per copy during the first and second half of the year. Rows represent regions, and columns represent magazines. \(G_1=\begin{pmatrix}1.20&1.50\\1.10&1.40\end{pmatrix}\), \(G_2=\begin{pmatrix}1.30&1.60\\1.00&1.50\end{pmatrix}\). In each half-year, every region sells \(10{,}000\) copies of \(Z_1\) and \(8000\) copies of \(Z_2\), so the sales vector for both periods is \(\mathbf{v}=\begin{pmatrix}10{,}000\\8000\end{pmatrix}\). Find the total annual profit in each region. Verify \((G_1+G_2)\mathbf{v}=G_1\mathbf{v}+G_2\mathbf{v}\), and interpret the matrix \(G_{\text{year}}=G_1+G_2\) in context.

Consider multistep processes modeled by \(\mathbf{v}_{n+1}=M\mathbf{v}_n\), where \(M\) is a \(2\times2\) matrix. 1. Let \(A=\begin{pmatrix}0.7&0.4\\0.3&0.6\end{pmatrix}\). Find a matrix \(Z\) such that \(A+Z=A\). 2. Show by multiplication that \(AZ\ne A\). 3. The multiplicative identity is the identity matrix \(I\). Describe what happens to the process when \(M=I\), and compare it with the theoretical case \(M=Z\).

Let \(A=\begin{pmatrix}1&2\\2&1\end{pmatrix}\). Find all real matrices \(B=\begin{pmatrix}a&b\\c&d\end{pmatrix}\) that commute with \(A\); that is, find all \(B\) such that \(AB=BA\).

A workshop produces two designer furniture pieces, \(F_1\) and \(F_2\), from three raw materials, \(M_1\), \(M_2\), and \(M_3\), through two intermediate production stages. The material flow is represented by \(A=\begin{pmatrix}1&0\\2&2\\0&3\end{pmatrix}\), which gives raw materials per assembly; \(B=\begin{pmatrix}2&1&0\\1&0&2\end{pmatrix}\), which gives assemblies per module; and \(C=\begin{pmatrix}1&2\\3&0\\0&1\end{pmatrix}\), which gives modules per furniture piece. Find the total-requirement matrix \(G\), whose entries give raw-material units per furniture piece. Then find the total number of units of raw material \(M_2\) needed for an order of \(10\) units of \(F_1\) and \(5\) units of \(F_2\).

Let \(M=\begin{pmatrix}2&0&1\\1&3&0\\0&1&2\end{pmatrix}\) and \(D=\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\). Find \(MD\) and \(DM\), and compare the results. What special effect does multiplication by a diagonal matrix have?

A dynamic system is defined by \(x_{n+1}=x_n+y_n\) and \(y_{n+1}=x_n\), with \(x_0=1\) and \(y_0=0\). a) Write the system in the form \(\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}=A\begin{pmatrix}x_n\\y_n\end{pmatrix}\), and give the matrix \(A\). b) Find the first six terms \(x_0\) through \(x_5\). Identify the sequence. c) Use repeated matrix multiplication to find \(x_6\), \(x_7\), and \(x_8\). Then calculate \(\frac{x_5}{x_4}\), \(\frac{x_6}{x_5}\), \(\frac{x_7}{x_6}\), and \(\frac{x_8}{x_7}\). Compare the ratios with \(\frac{1+\sqrt{5}}{2}\approx1.618\).

A commercial bakery makes three breads from wheat, rye, and spelt flour. The flour percentages are shown below. <table> <tr> <td></td> <th colspan="3">Flour percentage</th> </tr> <tr> <th>Bread</th> <th>Wheat</th> <th>Rye</th> <th>Spelt</th> </tr> <tr> <td>Harvest Wheat</td> <td>\(50\%\)</td> <td>\(40\%\)</td> <td>\(10\%\)</td> </tr> <tr> <td>Country Rye</td> <td>\(10\%\)</td> <td>\(80\%\)</td> <td>\(10\%\)</td> </tr> <tr> <td>Spelt Loaf</td> <td>\(5\%\)</td> <td>\(5\%\)</td> <td>\(90\%\)</td> </tr> </table> Two stores place the following weekend orders: Store A: \(120\) Harvest Wheat loaves, \(80\) Country Rye loaves, and \(50\) Spelt Loaves. Store B: \(150\) Harvest Wheat loaves, \(100\) Country Rye loaves, and \(40\) Spelt Loaves. Each loaf uses \(1.5\,\text{lb}\) of flour mixture. Use matrices to find the total pounds of each flour required.

An electronics manufacturer produces two control modules: audio modules, \(A\), and video modules, \(V\). The modules are made from resistors, capacitors, and transistors. The component requirements per module are \(M_{CM}=\begin{pmatrix}10&15\\5&10\\2&5\end{pmatrix}\), where the rows represent resistors, capacitors, and transistors, and the columns represent audio and video modules. Radios and televisions are assembled from the modules. The module requirements per finished product are \(M_{MP}=\begin{pmatrix}2&1\\0&3\end{pmatrix}\), where the columns represent radios and televisions. A wholesaler orders \(50\) radios and \(30\) televisions. a) Find the matrix \(M_{CP}\) that gives component requirements per finished product. b) Find the total numbers of resistors, capacitors, and transistors needed for the order.

The line through the origin \(g:\mathbf{x}=\lambda\begin{pmatrix}3\\1\end{pmatrix}\), where \(\lambda\in\mathbb R\), is a line of reflection. a) Find the reflection of \(P=(-1, 3)\) across \(g\). b) The reflection is represented by a matrix \(M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). Determine \(M\) by setting up and solving linear equations based on the images of a vector on \(g\) and the vector for \(P\).

The matrix \(A=\begin{pmatrix}-0.6&0.8\\0.8&0.6\end{pmatrix}\) represents reflection across a line \(h\) through the origin. a) Verify that applying the reflection twice gives the identity transformation by showing that \(A^2=I\). b) The reflection line consists of the fixed points of the transformation. Find an equation of \(h\) by solving \((A-I)\mathbf{x}=\mathbf{0}\).

A four-state process is represented by a matrix \(M\). The states are cyclically relabeled so that old state 1 becomes new state 2, old state 2 becomes new state 3, old state 3 becomes new state 4, and old state 4 becomes new state 1. The relabeled matrix is \(M^{\prime}=SMS^T\), where \(S\) permutes rows and \(S^T\) permutes columns. a) Find \(S\). b) Find \(M^{\prime}\) when \(M=\begin{pmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\end{pmatrix}\).

A catering company creates three weekly meal plans, \(W_1\), \(W_2\), and \(W_3\), in several production stages. Three basic ingredients, \(I_1\), \(I_2\), and \(I_3\), are used to prepare two meals, \(M_1\) and \(M_2\). The matrix \(B=\begin{pmatrix}2&1\\1&2\\3&1\end{pmatrix}\) gives the number of ingredient units needed for each meal. The ingredients contain two nutrients, \(N_1\) and \(N_2\). The matrix \(A=\begin{pmatrix}12&15&10\\8&5&14\end{pmatrix}\) gives the nutrient units in one unit of each ingredient. The numbers of each meal included in the weekly plans are \(C=\begin{pmatrix}100&50&80\\40&120&60\end{pmatrix}\). Find the total nutrient requirements for the three weekly plans in two ways: 1. First find the nutrient content of each meal, and then find the requirements for the weekly plans. 2. First find the total ingredient requirements for the weekly plans, and then find the nutrient requirements. Compare the results and identify the matrix property illustrated.

An electronics manufacturer produces three devices, \(G_1\), \(G_2\), and \(G_3\), from two modules, \(K_1\) and \(K_2\). The module requirements per device are \(M_2=\begin{pmatrix}3&1&2\\2&4&0\end{pmatrix}\). Two raw materials, \(R_1\) and \(R_2\), are needed to make the modules. Their requirements per module are \(M_1=\begin{pmatrix}4&2\\1&5\end{pmatrix}\). Two large orders, \(O_1\) and \(O_2\), are represented by \(D=\begin{pmatrix}10&5\\20&15\\15&10\end{pmatrix}\). a) Find \(M_{\text{total}}=M_1M_2\), and explain what its entries mean in context. b) Find the total raw-material requirements for both orders. Use the associative property and first calculate the module requirements for each order.