This is the perfect puzzle to anyone who never has solved a logic grid puzzle. Print full size. Of the value of the manufacturing produced, .25y pays for its internal energy and .10y pays for manufacturing consumed internally. Let’s do this “by hand”: \(\displaystyle {{P}^{2}}=\,\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\,=\left[ {\begin{array}{*{20}{c}} {(4\times 4)+(-6\times -2)} & {(4\times -6)+(-6\times 8)} \\ {(-2\times 4)+(8\times -2)} & {(-2\times -6)+(8\times 8)} \end{array}} \right]\,=\,\left[ {\begin{array}{*{20}{c}} {28} & {-72} \\ {-24} & {76} \end{array}} \right]\). IXL will track your score, and the questions will … Thus we could see that we read 6 paper fiction, 9 online fiction, 6 paper non-fiction, 5 online non-fiction books, and 13 paper and 14 online magazines. 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 sum of its last two digits is equal to its second digit increased by 5, the sum of its outer digits equals to its second digit decreased by 3. The files are grouped by difficulty (very easy, easy and medium) and are a great activity for all ages. You want to keep track of how many different types of books and magazines you read, and store that information in matrices. Thus, \(\displaystyle {{D}_{x}}=\det \left[ {\begin{array}{*{20}{c}} {\boldsymbol{{15}}} & 3 & {-1} \\ {\boldsymbol{{19}}} & {-3} & {-1} \\ {\boldsymbol{{-4}}} & {-3} & 3 \end{array}} \right]=-270\). This way we get rid of the number of cups of Almonds, Cashews, and Pecans, which we don’t need. Let’s look at the question that is being asked and define our variables: Let \(r=\) the number of roses, \(t=\) the number of tulips, and \(l=\) the number of lilies. Word problems Here is a list of all of the skills that cover word problems! Here are a couple more types of matrices problems you might see: Let \(P=\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\). What we really should have done with this problem is to use matrix multiplication separately for each girl; for example, for Brielle, we should have multiplied \(\left[ {\begin{array}{*{20}{c}} {16.4} & {19} & {17.5} \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} 2 \\ 1 \\ 2 \end{array}} \right]=\left[ {86.8} \right]\) and so on. And we did a 2 … We can come up with the following matrix multiplication: \(\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Junior}\,\,\,\,\,\,\text{Senior}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{H}\,\,\,\,\,\,\,\,\,\text{S}\,\,\,\,\,\,\,\,\text{C}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{H}\,\,\,\,\,\,\,\text{S}\,\,\,\text{ }\,\,\,\,\text{C}\\\begin{array}{*{20}{c}} {\text{Male}} \\ {\text{Female}} \end{array}\,\,\,\left[ {\begin{array}{*{20}{c}} {100} & {80} \\ {120} & {100} \end{array}} \right]\,\,\times \,\begin{array}{*{20}{c}} {\text{Junior}} \\ {\text{Senior}} \end{array}\,\,\,\left[ {\begin{array}{*{20}{c}} {.15} & {.35} & {.50} \\ {.25} & {.30} & {.45} \end{array}} \right]\,\,=\,\,\left[ {\begin{array}{*{20}{c}} {35} & {59} & {86} \\ {43} & {72} & {105} \end{array}} \right]\begin{array}{*{20}{c}} {\text{Male}} \\ {\,\,\,\,\,\,\text{Female}} \end{array}\end{array}\). Let’s use our calculator to put \(P\) in \([A]\) and \(\displaystyle \left[ {\begin{array}{*{20}{c}} 5 \\ 0 \end{array}} \right]\) in \([B]\). Soon we will be solving Systems of Equations using matrices, but we need to learn a few mechanics first! (c) Since \(\displaystyle \,\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\,\times \,Q=\left[ {\begin{array}{*{20}{c}} 5 \\ 0 \end{array}} \right]\), we have \(\displaystyle Q=\,{{\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]}^{{-1}}}\times \,\left[ {\begin{array}{*{20}{c}} 5 \\ 0 \end{array}} \right]\) (sort of like when we’re solving a system). Here is one: An outbreak of Chicken Pox hit the local public schools. As an example, if you had three sisters, and you wanted an easy way to store their age and number of pairs of shoes, you could store this information in a matrix. Percentage word problems Profit and loss word problems Markup and markdown word problems Decimal word problems. It turns out that we have extraneous information in this matrix; we only need the information where the girls’ names line up. Ratio and proportion word problems. But, like we learned in the Systems of Linear Equations and Word Problems Section here, sometimes we have systems where we either have no solutions or an infinite number of solutions. If we mix 180 g of water from the first container with 120 g of water from the second container, the resulting water temperature will be 46Â°C. For example, to find out how many healthy males we would have, we’d set up the following equation and do the calculation: \(.15(100)+.25(80)=35\). Matrix Logic Corporation (MLC) provides Content and Document Management resources throughout the United States and Canada. 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. When you multiply a square matrix with an identity matrix, you just get that matrix back: \(\displaystyle \left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]\,\times \,\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]\). Note again that only square matrices have inverses, but there are square matrices that don’t have one (when the determinant is 0): \(\displaystyle \text{Inverse }\left[ {\begin{array}{*{20}{c}} {{{a}_{{11}}}} & {{{a}_{{12}}}} \\ {{{a}_{{21}}}} & {{{a}_{{22}}}} \end{array}} \right]=\frac{1}{{\det A}}\left[ {\begin{array}{*{20}{c}} {{{a}_{{22}}}} & {-{{a}_{{12}}}} \\ {-{{a}_{{21}}}} & {{{a}_{{11}}}} \end{array}} \right]\), \(\displaystyle \color{#800000}{{\text{Inverse }\left[ {\begin{array}{*{20}{c}} 3 & 1 \\ 4 & 8 \end{array}} \right]}}=\frac{1}{{20}}\left[ {\begin{array}{*{20}{c}} 8 & {-1} \\ {-4} & 3 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{2}{5}} & {-\frac{1}{{20}}} \\ {-\frac{1}{5}} & {\frac{3}{{20}}} \end{array}} \right]\), \(\displaystyle \color{#800000}{{\text{Inverse }\left[ {\begin{array}{*{20}{c}} 3 & 6 \\ 2 & 4 \end{array}} \right]}}=\frac{1}{0}\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 3 \end{array}} \right]=\text{No Inverse}\). Think of an identity matrix like “1” in regular multiplication (the multiplicative identity), and the inverse matrix like a reciprocal (the multiplicative inverse). Note that, like the other systems, we can do this for any system where we have the same numbers of equations as unknowns. Basic 3. Then we’ll “divide” by the matrix in front of \(X\). \(\displaystyle \begin{align}\left[ {\begin{array}{*{20}{c}} 2 & 3 \\ 1 & {-4} \end{array}} \right]\,X-\,\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]&=\,\left[ {\begin{array}{*{20}{c}} 5 & 0 \\ {-2} & 3 \end{array}} \right]\,+\left[ {\begin{array}{*{20}{c}} 4 & {-6} \\ {-2} & 8 \end{array}} \right]\\\,\,\,\left[ {\begin{array}{*{20}{c}} 2 & 3 \\ 1 & {-4} \end{array}} \right]\,X&=\,\left[ {\begin{array}{*{20}{c}} 9 & {-6} \\ {-4} & {11} \end{array}} \right]\\X&={{\left[ {\begin{array}{*{20}{c}} 2 & 3 \\ 1 & {-4} \end{array}} \right]}^{{-1}}}\,\left[ {\begin{array}{*{20}{c}} 9 & {-6} \\ {-4} & {11} \end{array}} \right]\,=\,\left[ {\begin{array}{*{20}{c}} {\frac{{24}}{{11}}} & {\frac{9}{{11}}} \\ {\frac{{17}}{{11}}} & {-\frac{{28}}{{11}}} \end{array}} \right]\end{align}\). 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