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## matrix logic word problems

After you’ve stored the square matrix, hit , and hitonce so that MATH is highlighted. Put 3 by 4 matrix $$\displaystyle \left[ {\begin{array}{*{20}{c}} 5 & {-6} & {-7} & 7 \\ 6 & {-4} & {10} & {-34} \\ 2 & 4 & {-3} & {29} \end{array}} \right]$$ in [A]. We’ll learn other ways to use the calculator with matrices a little later. Problem 27. It’s really not too difficult; it can just be a lot of work, so again, I’ll take the liberty of using the calculator to do most of the work , Let’s just show an example; let’s solve the following system using Cramer’s rule:  $$\displaystyle \begin{array}{l}\,2x+3y-\,\,z\,=\,15\\4x-3y-\,\,z\,=\,19\\\,\,x\,-\,3y+\,3z\,=\,-4\end{array}$$. Here’s an example of matrices with dimensions that would work: Notice how the “middle” or “inner” dimensions of the first matrices have to be the same (in this case, “2”), and the new matrix has the “outside” or “outer” dimensions of the first two matrices (“3 by 5”). Let’s look at a matrix that contains numbers and see how we can add and subtract matrices. Note that the determinant of a matrix can be designated by $$\det \left[ \text{A} \right]$$  or  $$\left| \text{A} \right|$$, and the inverse of a matrix by  $${{\text{A}}^{{-1}}}$$. A typical propositional logic word problem is as follows:. (a) Find $$2P$$,    (b) Find $${{P}^{2}}$$,   (c) Find $$Q$$ when $$P\times Q=\left[ {\begin{array}{*{20}{c}} 5 \\ 0 \end{array}} \right]$$. Solve the matrix word problems on Math-Exercises.com - Collection of math problems & math exercises. If the third dimension of the cuboid increases by 3 cm, its surface area increases by 126 cm2. Oh well, no harm done; and now you’ll know what to do if you see these types of matrices problems. Thus, $$\displaystyle {{D}_{x}}=\det \left[ {\begin{array}{*{20}{c}} {\boldsymbol{{15}}} & 3 & {-1} \\ {\boldsymbol{{19}}} & {-3} & {-1} \\ {\boldsymbol{{-4}}} & {-3} & 3 \end{array}} \right]=-270$$. Use your logic to figure out who did what in each of these wacky scenarios. You want to keep track of how many different types of books and magazines you read, and store that information in matrices. To get the $$x, y$$, and $$z$$ answers to the system, you simply divide the determinants $${{D}_{x}}$$, $${{D}_{y}}$$, and $${{D}_{z}}$$, by the determinant $$D$$, respectively. Then type, and hit ENTER for matrix [A], or scroll to the matrix you want. (b)  When we square P, we just multiply it by itself. You may have heard matrices called arrays, especially in computer science. You want to keep track of how many different types of books and magazines you read, and store that information in matrices. 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. (It is important to note that if we are trying to solve a system of equations and the determinant turns out to be 0, that system either has an infinite number of solutions, or no solution.). Logic puzzles are a great way for kids to work on critical problem solving skills that help not just with puzzle solving, but with standardized testing as well. First of all, you can only multiply matrices if the dimensions “match”; the second dimension (columns) of the first matrix has to match the first dimension (rows) of the second matrix, or you can’t multiply them. Pretty clever! The actual matrix is inside and includes the brackets: $$\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {\text{Ashley}} & {\text{Emma}} & {\text{Chloe}} \end{array}\\\begin{array}{*{20}{c}} {\text{Age}} \\ {\text{Number of Pairs of Shoes}} \end{array}\text{ }\left[ {\begin{array}{*{20}{c}} {\text{23}} & {\,\,\,\,\,\,\,\,\,\text{18}} & {\,\,\,\,\,\,\,\,\,\text{15}} \\ \text{5} & {\,\,\,\,\,\,\,\,\text{23}} & {\,\,\,\,\,\,\,\,\,\text{12}} \end{array}} \right]\end{array}$$. 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). Multiply each of the top numbers by the determinant of the 2 by 2 matrix that you get by crossing out the other numbers in that top number’s row and column. Cool Math and Logic for Kids – At this site, elementary-age kids will find puzzles, word problems, and animated logic problems. In each puzzle you are given a series of categories, and an equal number of options within each category. There are twice as many women as men and four times as many children as women. Think of it like the inner dimensions have to match, and the resulting dimensions of the new matrix are the outer dimensions. Students must define, give an example, write a word problem,and then explain how to solve that word problem, for the identy, distributive, associative, and communicative multiplication properties. (a)   If the capacity of energy production is $15 million and the capacity of manufacturing production is$20 million, how much of each is consumed internally for capacity production? It makes sense to put the first group of data into a matrix with Almonds, Cashews, and Pecans as columns, and then put the second group of data into a matrix with information about Almonds, Cashews, and Pecans as rows. We take the positive only since the determinant is positive. Word problems on mixed fractrions. The dimensions of this matrix are  “2 x 3” or “2  by 3”, since we have 2 rows and 3 columns. Watch the order when we multiply by the inverse (matrix multiplication is not commutative), and thank goodness for the calculator! Let’s say you’re in avid reader, and in June, July, and August you read fiction and non-fiction books, and magazines, both in paper copies and online. A florist is making 5 identical bridesmaid bouquets for a wedding. You can hit MATH ENTER (for Frac) to get the matrix in fractions: Note that a matrix, multiplied by its inverse, if it’s defined, will always result in what we call an Identity Matrix:  $$\displaystyle \left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} {\frac{2}{5}} & {-\frac{1}{{20}}} \\ {-\frac{1}{5}} & {\frac{3}{{20}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right]$$. Here are the three equations: $$\begin{array}{c}x+y+z=26\\z=2y\\z=3x-1\end{array}$$. Let’s call this first determinant $$D$$;  $$\displaystyle D=\det \left[ {\begin{array}{*{20}{c}} 2 & 3 & {-1} \\ 4 & {-3} & {-1} \\ 1 & {-3} & 3 \end{array}} \right]=-54$$. One row of the coefficient matrix (and the corresponding constant matrix) is a multiple of another row. From counting through calculus, making math make sense! 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. A system that has an infinite number of solutions may look like this: \displaystyle \begin{align}2x+2y-\,z\,&=16\\4x+4y-2z&=32\\\,\,x\,-\,3y+3z&=-4\end{align}. It works! Here is that information, and how it would look in matrix form: Matrix Form:  $$\left[ {\begin{array}{*{20}{c}} 2 & 4 \\ \begin{array}{l}3\\4\end{array} & \begin{array}{l}1\\5\end{array} \end{array}} \right]$$, Matrix Form:  $$\left[ {\begin{array}{*{20}{c}} 3 & 2 \\ \begin{array}{l}1\\5\end{array} & \begin{array}{l}1\\3\end{array} \end{array}} \right]$$, Matrix Form:  $$\left[ {\begin{array}{*{20}{c}} 1 & 3 \\ \begin{array}{l}2\\4\end{array} & \begin{array}{l}3\\6\end{array} \end{array}} \right]$$. The sulfuric acid consists of hydrogen, oxygen and sulfur, wherein the weight ratio of the hydrogen and the sulfur is 1:16 and the weight ratio of the oxygen and the sulfur is 2:1. 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.). \displaystyle \begin{align}\pm \frac{1}{2}\left| {\begin{array}{*{20}{c}} {{{a}_{1}}} & {{{b}_{1}}} & 1 \\ {{{a}_{2}}} & {{{b}_{2}}} & 1 \\ {{{a}_{3}}} & {{{b}_{3}}} & 1 \end{array}} \right|&=\pm \frac{1}{2}\left| {\begin{array}{*{20}{c}} {-1} & 3 & 1 \\ 0 & {-5} & 1 \\ 2 & 8 & 1 \end{array}} \right|=\pm \frac{1}{2}\left[ {\left( {-1} \right)\left( {-5\cdot 1-1\cdot 8} \right)-3\left( {0\cdot 1-1\cdot 2} \right)+1\left( {0\cdot 8–5\cdot 2} \right)} \right]\\&=\pm \frac{1}{2}\left( {29} \right)=\frac{1}{2}\left( {29} \right)=14.5\end{align}. (% is meant as by volume). Soon we will be solving Systems of Equations using matrices, but we need to learn a few mechanics first! Without going too much into Geometry, let’s look at what it looks like when three systems (each system looks like a “plane” or a piece of paper) have an infinite number of solutions, no solutions, and one solution, respectively: eval(ez_write_tag([[336,280],'shelovesmath_com-leader-3','ezslot_6',134,'0','0']));Systems that have an infinite number of solutions (called dependent or coincident) will have two equations that are basically the same. Hit ENTER or 1 for det(. It turns out that we have extraneous information in this matrix; we only need the information where the girls’ names line up. From jigsaw puzzles to acrostics, logic puzzles to drop quotes, patchwords to wordtwist and even sudoku and crossword puzzles, we run the gamut in word puzzles, printable puzzles and logic … The TI graphing calculator is great for matrix operations! This is the currently selected item. Her supplier has provided the following nutrition information: Her first mixture, Mixture 1, consists of 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. $$\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Outputs:}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Energy}\,\,\,\,\,\,\,\text{Manufacturing}\\\,\,\,\,\text{Inputs}:\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {\text{Energy }} \\ {\text{Manufacturing }} \end{array}\,\,\,\,\,\left[ {\begin{array}{*{20}{c}} {.40} & {\,\,\,\,\,\,\,\,.25} \\ {.25} & {\,\,\,\,\,\,\,\,\,.10} \end{array}} \right]\end{array}$$. Logic puzzles appear reguarly on standardized tests, and when kids practice with various types of logic puzzles they are better prepared when unfamiliar puzzles show up. If the second dimension of the cuboid increases by 2 cm, the surface area of the cuboid increases by 96 cm2. Printable Word Problems – Kids in grades K- 12 will find something of interest here. In the last video we saw how a matrix and figuring out its inverse can be used to solve a system of equations. Let’s put the money terms together, and also the counting terms together: $$\begin{array}{l}6r+4t+3l=610\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(price of each flower times number of each flower = total price)}\\r=2(t+l)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(two times the sum of the other two flowers = number of roses)}\\r+t+l=5(24)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(total flowers = 5 bouquets, each with 24 flowers)}\end{array}$$. We can express the amounts (proportions) the industries consume in matrices, such as in the following problem: The following coefficient matrix, or input-output matrix, shows the values of energy and manufacturing consumed internally needed to produce $1 of energy and manufacturing, respectively. For example, to find out how many healthy males we would have, we’d set up the following equation and do the calculation: $$.15(100)+.25(80)=35$$. If one dimension of the cuboid increases by 1 cm, the surface area of the cuboid increases by 54 cm2. Note again that only square matrices have inverses, but there are square matrices that don’t have one (when the determinant is 0): $$\displaystyle \text{Inverse }\left[ {\begin{array}{*{20}{c}} {{{a}_{{11}}}} & {{{a}_{{12}}}} \\ {{{a}_{{21}}}} & {{{a}_{{22}}}} \end{array}} \right]=\frac{1}{{\det A}}\left[ {\begin{array}{*{20}{c}} {{{a}_{{22}}}} & {-{{a}_{{12}}}} \\ {-{{a}_{{21}}}} & {{{a}_{{11}}}} \end{array}} \right]$$, $$\displaystyle \color{#800000}{{\text{Inverse }\left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]}}=\frac{1}{{20}}\left[ {\begin{array}{*{20}{c}} 8 & {-1} \\ {-4} & 3 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{2}{5}} & {-\frac{1}{{20}}} \\ {-\frac{1}{5}} & {\frac{3}{{20}}} \end{array}} \right]$$, $$\displaystyle \color{#800000}{{\text{Inverse }\left[ {\begin{array}{*{20}{c}} 3 & 6 \\ 2 & 4 \end{array}} \right]}}=\frac{1}{0}\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 3 \end{array}} \right]=\text{No Inverse}$$. 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Word puzzles math games to entertain you for hours the multiplication works States. ), or scroll to the matrix you want to overthink the Solutions a chart or matrix often help solving... S look at a matrix set up the matrix is not square or! Matrices, but you do n't want to keep track of how many kilograms of zinc is kg/m3. Price of things at two supermarkets are different in different cities Winks is a multiple matrix logic word problems another row series. Then type, and practice, practice, practice New matrix are the amounts produced 5,500 men women... Design 5th edition by Charles H. … from counting through calculus, math! Left corner, multiply diagonally down and add those three products ( moving to matrix.