WIR Math 166-copyright Joe Kahlig, 09C Page 1Week in Review–Additional Chapter 5 MaterialSection 5.2: Matrix Multiplication• if A is a mxn matrix and B is a nxp matrix then the matrix AB has a size of mxp• note: the number of columns of A equals the number of rows of B.• identity matrix In• size nxn• all zeros except for a1,1= a2,2= a3,3= ... = 1A ="7 2 46 5 0#B ="9 3 0−1 2 8#C =1 3−2 52 0D ="x 12 5#1. Use the above matrices to compute the following.(a) AC =(b) BD =(c) DB =(d) D2=(e) 21 0 00 1 00 0 11 4 82 3 00 1 5=(f)1 0 00 1 00 0 11 4 82 3 00 1 58 2 5=2. A dietitian plans a meal a r ound two foods. The number of units of vitamin A andvitamin C in each ounce of these foods is represented by the matrix M.Food I Food II Food I Food II Food I Food IIM =Vitamin AVitamin C"30 907 45#B =h5 2iL =h9 4iThe matrices B and L represent t he amount of each food (in ounces) consumed by thegirl at breakfast and lunch, respectively. Explain the meaning of the entries in thesecomputations..(a) LM =h298 990i(b) MBT="330125#WIR Math 166-copyright Joe Kahlig, 09C Pa ge 2Section 5.3: The inverse of a Matrix• the matrix must be square.• NOTall square matrices have an inverse.• the inverse of A is denoted A−1• AA−1= A−1A = I• A system of equations may be written as a matrix equation: AX=B• A is the coefficient matrix• X is the variable matrix• If A has an inverse, then the solution is X = A−1B.• Matrix A not having an inverse does not imply that the system of equations has nosolution. It means that you need to try another method to solve the problem.3. If A ="5 15 2#find A−14. If A =2 1 13 2 12 1 2find A−15. If A =2 1 14 2 32 1 2find A−16. Determin if the matrices A and B are inverses of each other.A =1 2 x0 5 20 2 1B =1 2x − 2 4 − 5x0 1 −20 −2 57. Answer the following using this system of equations.2x + z = 22x + y − z = 13x + y − z = 4(a) Write down the coefficient matrix.(b) Write the system of equations as a matrix equation.(c) Solve the system of equations using matrices.8. True or False. A system of equations is represented by the matrix equation AX = B.If the coefficient matrix, A, does not have an inverse then the system of equations do esnot have a solution.9. Solve for the matrix X. Assume that all matrices are square and all needed inversesare possible.(a) BX = E − CX(b) XJ + XA =
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