Matrix word problem: vector combination. Sometimes you’ll have to learn Cramer’s Rule, which is another way to solve systems with matrices. We also know that tests are 40% of the grade, projects 15%, homework 25%, and quizzes 20%. You can probably guess what the next determinant we need is: $${{D}_{y}}$$, which we get by “throwing away” the second column ($$y$$) of the original matrix and replacing the numbers with the constant column like we did earlier for the $$x$$. The following matrix consists of a shoe store’s inventory of flip flops, clogs, and Mary Janes in sizes small, medium, and large: The store wants to know how much their inventory is worth for all the shoes. The activities include visual puzzles, logic problems, multi-step problems, and more. If the third dimension of the cuboid increases by 3 cm, its surface area increases by 126 cm2. Word problems on mixed fractrions. Much easier than figuring it out by hand! You will probably find this puzzle as easy as the first basic one. Using two matrices and one matrix equation, find out how many males and how many females (don’t need to divide by class) are healthy, sick, and carriers. There will be 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females. Print full size. A Hadamard matrix is an n nmatrix H with entries in f 1;+1gsuch that any two distinct rows or columns of Hhave inner product 0. LOGIC PUZZLES. Over 2,000 math exercises. Determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures. In your Geometry class, you may learn a neat trick where we can get the area of a triangle using the determinant of a matrix. OK, now for the fun and easy part! If we write the digits of the number in reverse order, the number increases by 2,178. This is the currently selected item. Here are some basic steps for storing, multiplying, adding, and subtracting matrices: $$\color{#800000}{{\left[ {\begin{array}{*{20}{c}} 2 & {-1} \\ 3 & 2 \\ 7 & 5 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} 0 & {-4} & 3 & 1 & 4 \\ 6 & 7 & 2 & 9 & {-3} \end{array}} \right]\,\,}}\,=\,\,\left[ {\begin{array}{*{20}{c}} {-6} & {-15} & 4 & {-7} & {11} \\ {12} & 2 & {13} & {21} & 6 \\ {30} & 7 & {31} & {52} & {13} \end{array}} \right]$$, (Note that you can also enter matrices using ALPHA ZOOM and the arrow keys in the newer graphing calculators.). The numbers in bold are our answers: Sometimes you’ll get a matrix word problem where just numbers are given; these are pretty tricky. In general, an m n matrix has m rows and n columns and has mn entries. This is the perfect puzzle to anyone who never has solved a logic grid puzzle. This way we get rid of the number of cups of Almonds, Cashews, and Pecans, which we don’t need. A chart or matrix often help in solving problems. Think of an, ” in regular multiplication (the multiplicative identity), and the, (It is important to note that if we are trying to solve a system of equations and the determinant turns out to be, Solve the matrix equation for $$X$$ ($$X$$, $$\displaystyle \begin{array}{l}\,2x+3y-\,\,z\,=\,15\\4x-3y-\,\,z\,=\,19\\\,\,x\,-\,3y+\,3z\,=\,-4\end{array}$$, \displaystyle \begin{align}2x+2y-\,z\,&=16\\4x+4y-2z&=32\\\,\,x\,-\,3y+3z&=-4\end{align}, \displaystyle \begin{align}2x+2y-\,z\,&=16\\4x+4y-2z&=\,10\\\,\,x\,-\,3y+3z&=-4\end{align}, Her first mixture, Mixture 1, consists of, An outbreak of Chicken Pox hit the local public schools. But we have to be careful, since these amounts are for 10 cups (add down to see we’ll get 10 cups for each mixture in the second matrix above). (The density of copper is 8,900 kg/m3, the density of zinc is 7,100 kg/m3.). To solve systems with matrices, we use $$\displaystyle X={{A}^{{-1}}}B$$. Here’s a problem from the Systems of Linear Equations and Word Problems Section; we can see how much easier it is to solve with a matrix. eval(ez_write_tag([[468,60],'shelovesmath_com-medrectangle-3','ezslot_7',109,'0','0']));Matrices are called multi-dimensional since we have data being stored in different directions in a grid. By adding 30 kg of pure tin we have to prepare a bronze alloy made out of 75% of copper and 25% of tin. There are 5,500 men, women and children altogether at the swimming pool. But since we know that we have both juniors and seniors with males and females, the first matrix will probably be a 2 x 2. Voiceover:The price of things at two supermarkets are different in different cities. Here are some examples of those applications. Solve these word problems, with answers included. We’ll learn other ways to use the calculator with matrices a little later. Find the number. (It doesn’t matter which side; just watch for negatives). This lesson will show you an example of using matrix logic to exhaust possibilities until the solution becomes evident. The files are grouped by difficulty (very easy, easy and medium) and are a great activity for all ages. This one’s a little trickier, since it doesn’t really look like a systems problem, but you solve it the same way: Solve the matrix equation for $$X$$ ($$X$$ will be a matrix): $$\displaystyle \left[ {\begin{array}{*{20}{c}} 2 & 3 \\ 1 & {-4} \end{array}} \right]X-\,\,\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\,=\,\left[ {\begin{array}{*{20}{c}} 5 & 0 \\ {-2} & 3 \end{array}} \right]$$. We can come up with the following matrix multiplication: $$\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Junior}\,\,\,\,\,\,\text{Senior}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{H}\,\,\,\,\,\,\,\,\,\text{S}\,\,\,\,\,\,\,\,\text{C}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{H}\,\,\,\,\,\,\,\text{S}\,\,\,\text{ }\,\,\,\,\text{C}\\\begin{array}{*{20}{c}} {\text{Male}} \\ {\text{Female}} \end{array}\,\,\,\left[ {\begin{array}{*{20}{c}} {100} & {80} \\ {120} & {100} \end{array}} \right]\,\,\times \,\begin{array}{*{20}{c}} {\text{Junior}} \\ {\text{Senior}} \end{array}\,\,\,\left[ {\begin{array}{*{20}{c}} {.15} & {.35} & {.50} \\ {.25} & {.30} & {.45} \end{array}} \right]\,\,=\,\,\left[ {\begin{array}{*{20}{c}} {35} & {59} & {86} \\ {43} & {72} & {105} \end{array}} \right]\begin{array}{*{20}{c}} {\text{Male}} \\ {\,\,\,\,\,\,\text{Female}} \end{array}\end{array}$$. You can actually define the set of solutions by just allowing $$z$$ to be anything, and then, from the other rows, solve for $$x$$ and $$y$$ in terms of $$z$$: This would look like  \displaystyle \begin{align}1x+0y+.375z&=5\\0x+1y-.875z&=3\\\,\,\,\,\,\,\,\text{For all }z,\,\,\,\,\,\,0&=0\end{align}, so the solution set for $${x,y,z}$$ is $$\displaystyle \{5-.375z,5-.375z,z\}$$. Download and play for free our printable logic grid puzzles (PDF). Toilet paper in Duluth, Minnesota cost 3.99 a package while toilet paper in New York City cost 8.95 a package. Then, starting back from the upper right corner, multiply diagonally down and subtract those three products (moving to the left). You may have heard matrices called arrays, especially in computer science. Sometimes we can just put the information we have into matrices to sort of see what we are going to do from there. If the second dimension of the cuboid increases by 2 cm, the surface area of the cuboid increases by 96 cm2. A nut distributor wants to know the nutritional content of various mixtures of almonds, cashews, and pecans. The first table below show the points awarded by judges at a state fair for a crafts contest for Brielle, Brynn, and Briana. For example, for the matrix above, “Ashley’s number of pairs of shoes (5)” would be identified as $${{a}_{{2,1}}}$$, since it’s on the 2nd row and it’s the 1st entry. Video transcript. Each option is used once and only once. Notice how the percentages in the rows in the second matrix add up to 100%. The actual matrix is inside and includes the brackets: $$\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {\text{Ashley}} & {\text{Emma}} & {\text{Chloe}} \end{array}\\\begin{array}{*{20}{c}} {\text{Age}} \\ {\text{Number of Pairs of Shoes}} \end{array}\text{ }\left[ {\begin{array}{*{20}{c}} {\text{23}} & {\,\,\,\,\,\,\,\,\,\text{18}} & {\,\,\,\,\,\,\,\,\,\text{15}} \\ \text{5} & {\,\,\,\,\,\,\,\,\text{23}} & {\,\,\,\,\,\,\,\,\,\text{12}} \end{array}} \right]\end{array}$$. For example, if we wanted to know the total number of each type of book/magazine we read, we could add each of the elements to get the sum: $$\displaystyle \require{cancel} \color{#800000}{{\left[ {\begin{array}{*{20}{c}} 2 & 4 \\ \begin{array}{l}3\\4\end{array} & \begin{array}{l}1\\5\end{array} \end{array}} \right]\,\,+\,\,\left[ {\begin{array}{*{20}{c}} 3 & 2 \\ \begin{array}{l}1\\5\end{array} & \begin{array}{l}1\\3\end{array} \end{array}} \right]\,\,\,+\,\,\left[ {\begin{array}{*{20}{c}} 1 & 3 \\ \begin{array}{l}2\\4\end{array} & \begin{array}{l}3\\6\end{array} \end{array}} \right]}}\,\,=\,\,\left[ {\begin{array}{*{20}{c}} {2+3+1} & {4+2+3} \\ \begin{array}{l}3+1+2\\4+5+4\end{array} & \begin{array}{l}1+1+3\\5+3+6\end{array} \end{array}} \right]\,\,=\,\,\left[ {\begin{array}{*{20}{c}} 6 & 9 \\ 6 & 5 \\ {13} & {14} \end{array}} \right]$$. Think of it like the inner dimensions have to match, and the resulting dimensions of the new matrix are the outer dimensions. Let’s multiply the following matrix using the calculator: By definition, the inverse of a matrix is the reciprocal of the determinant, multiplied by a “, \displaystyle \begin{align}\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right]&={{\left[ {\begin{array}{*{20}{c}} 1 & 1 \\ {25} & {50} \end{array}} \right]}^{{-1}}}\times \,\,\,\,\left[ {\begin{array}{*{20}{c}} 6 \\ {200} \end{array}} \right]\\\,\,&=\,\frac{1}{{25\,}}\left[ {\begin{array}{*{20}{c}} {50} & {-1} \\ {-25} & 1 \end{array}} \right]\times \,\left[ {\begin{array}{*{20}{c}} 6 \\ {200} \end{array}} \right]\\\,\,\,&=\left[ {\begin{array}{*{20}{c}} 2 & {-\frac{1}{{25}}} \\ {-1} & {\frac{1}{{25}}} \end{array}} \right]\times \,\left[ {\begin{array}{*{20}{c}} 6 \\ {200} \end{array}} \right]\\\,\,\,&=\left[ {\begin{array}{*{20}{c}} {(2\times 6)+(-\frac{1}{{25}}\times 200)} \\ {(-1\times 6)+(\frac{1}{{25}}\times 200)} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right]\end{align}. Because we can solve systems with the inverse of a matrix, since the inverse is sort of like dividing to get the variables all by themselves on one side. (a)  When we multiply a matrix by a scalar (number), we just multiply all elements in the matrix by that number. Hmm….this is interesting; we end up with a matrix with the girls’s names as both rows and columns. If we wanted to see how many book and magazines we would have read in August if we had doubled what we actually read, we could multiply the August matrix by the number 2. When you try to these types of systems in your calculator (using matrices), you’ll get an error since the determinant of the coefficient matrix will be 0. Printable Word Problems – Kids in grades K- 12 will find something of interest here. The third number is twice the second, and is also 1 less than 3 times the first. You should end up with entries that correspond with the entries of each row in the first matrix. We also know that tests are, Note that a matrix, multiplied by its inverse, if it’s defined, will always result in what we call an, :  $$\displaystyle \left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} {\frac{2}{5}} & {-\frac{1}{{20}}} \\ {-\frac{1}{5}} & {\frac{3}{{20}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right]$$, When you multiply a square matrix with an identity matrix, you just get that matrix back: $$\displaystyle \left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]$$. Math and Logic – Math games, word problems, and logic puzzles will entertain you for hours. Of these wacky scenarios you have to divide each answer by 10 to get the is! 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