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====== Math 445 lab 3 ====== **Problem 1:** Compute the following products in your head or with pencil and paper, and then check your answers with Matlab. Note: one of the problems is not well-posed. **(a)** <latex> x = \left[\begin{array}{ccc} 4 & 3 & 8 \end{array} \right] </latex> and <latex> y = \left[\begin{array}{c} 2 \\ 1 \\ -4 \end{array} \right] </latex>. What is $xy$? **(b)** <latex> x = \left[\begin{array}{ccc} 9 & 1 & 0 \end{array} \right] </latex> and <latex> y = \left[\begin{array}{ccc} -5 & 6 2 \end{array} \right] </latex>. What is $xy$ **%%(c)%%** <latex> A = \left[ \begin{array}{cc} 3 & 1 \\ 2 & -7 \end{array} \right] </latex> and <latex> x = \left[ \begin{array}{cc} 2 \\ 1 \end{array} \right] </latex>. What is $Ax$? **(d)** <latex> A = \left[ \begin{array}{cc} 3 & 1 \\ 2 & -7 \end{array} \right] </latex> and <latex> B = \left[ \begin{array}{cc} 4 & -6 \\ 0 & 2 \end{array} \right] </latex>. What is $AB$? **Problem 5:** In Matlab, create a 2 x 2 matrix <latex> B = \left[ \begin{array}{cc} 4 & 2 \\ -1 & 5 \end{array} \right] </latex> and a 2-d vector <latex> b = \left[ \begin{array}{cc} 3 \\ 4 \end{array} \right] </latex>. What vector $x$ satisfies $Ax=b$? **Problem 6:** Use Matlab to solve the problem. Nilanjana has 40 coins worth %%$%%6.40. They're all quarters and nickels. How many nickels and how many quarters does she have? Verify that your answer solves the problem. Hints: Convert the story problem into two equations in two unknowns. Then rewrite this system of equations in matrix-vector notation, $Ax=b$, where $A$ are the known coefficients of the linear equations, $x$ a vector of unknowns, and $b$ a vector of known constants. Enter the matrix $A$ and the known vector $b$ into Matlab, then solve for $x$ using Matlab's backslash operator: ''x = A\b''. **Problem 7:** Use Matlab to solve the problem. Suhasini has 44 coins worth %%$%%7.50. They're all quarter, dimes, and nickels. She has twice as many dimes as nickels. How many of each type of coin does she have? Find the answer, and then verify that the solution satisfies the problem.
