ࡱ> ?A>[ !bjbjmm 4jbb5k::    4 0$2i%i%i%Q0S0S0S0S0S0S0$25w0i%I# i%i%i%w0::G0=.=.=.i%::Q0=.i%Q0=.=.6=.P^ g(=.=0000=.h6{*h6=.h6U.i%i%=.i%i%i%i%i%w0w0q,i%i%i%0i%i%i%i%h6i%i%i%i%i%i%i%i%i%B : Accelerated Pre-Calculus Name________________________ Inverses and Matrix Equations Period _______Date ____________ I. Finding multiplicative inverses by hand: Using your calculator, find each of the following: 1.  EMBED Equation.DSMT4  ___ and  EMBED Equation.DSMT4  ________ 2.  EMBED Equation.DSMT4 _____ and  EMBED Equation.DSMT4 _________ 3  EMBED Equation.DSMT4  ___ and  EMBED Equation.DSMT4  ________ 4.  EMBED Equation.DSMT4 _____ and  EMBED Equation.DSMT4 _________ What patterns do you notice? II. Use those patterns and previous knowledge to find these: Find inverses for the following matrices by hand. Check #1, 2 and 3 with your calculator. 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  3.)  EMBED Equation.DSMT4  4.)  EMBED Equation.DSMT4  Given: A =  EMBED Equation.DSMT4  and B =  EMBED Equation.DSMT4 , 5.) Find det(A) 6.) Find |B|. 7.) Find A-1 8.) Find B-1 9.) Find A + B 10.) Find A ( B 11.) Find A ( B 12.) Can you find A ( B ?? 13.) Although it is impossible to divide matrices, remember that  EMBED Equation.DSMT4 , because  EMBED Equation.DSMT4  is the multiplicative inverse of 2. (Division is just multiplication by an inverse!) So if we want to divide matrix A by B, we can multiply A by Bs inverse. Do that and write it in fractional form. III. Multiplying by an inverse can be used to solve matrix equations. Consider the system of equations:  EMBED Equation.DSMT4  This can be represented by the matrix equation:  EMBED Equation.DSMT4  Think about solving a simple equation and compare it to solving a matrix equation.  Simple Equation: Matrix Equation: 4x = 12  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  x = 3 The solution is  EMBED Equation.DSMT4  Check your work using the calculator: Type in 2(8/5) + (14/5) = The result is 6. Furthermore, 3(8/5) (14/5) = yields 2. Set up and solve the matrix equations for these systems of equations. Then check your answers.  14.) 5x + 7y = 2 15.) 57x ( 51y = 129 16.) (25.2x + 64.8y = 21.9 8x ( 2y = 9 76x + 98y = 623 24.8x ( 14.4y = 73.6 Matrix equations are helpful in solving larger systems of equations also. Use the same method on these larger systems. Set up, solve and check matrix equations for these. Use your calculators to find the inverses.  x + y + z = 9 3x + 4y ( z = (10 7x + 5y ( 6z = (9 17.) 2x ( y + z = (9 18.) 5x ( 2y + 7z = 44 19.) 2x + 3y + 7z = 5 4x + y + 3z = 17 2x + y + 5z = 13 5x ( 7y + 9z = (6 Accelerated Pre-Calculus Name________________________ Inverses and Matrix Equations Period _______Date ____________ I. Finding multiplicative inverses by hand: Using your calculator, find each of the following: 1.  EMBED Equation.DSMT4 = _1_ and  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4 =_4_ and  EMBED Equation.DSMT4  3  EMBED Equation.DSMT4 = _-3_ and  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4 =_0_ and  EMBED Equation.DSMT4 undefined What patterns do you notice? The numbers on the main diagonal switch places. The other 2 numbers change signs and all numbers are divided by the determinant. II. Use those patterns and previous knowledge to find these: Find inverses for the following matrices by hand. Check #1, 2 and 3 with your calculator.  1. EMBED Equation.DSMT4  2. EMBED Equation.DSMT4  3. EMBED Equation.DSMT4  4. EMBED Equation.DSMT4  Given: A =  EMBED Equation.DSMT4  and B =  EMBED Equation.DSMT4 , 5.) Find det(A) 6.) Find |B|. 7.) Find A-1  EMBED Equation.DSMT4  -6 63 = -69 24 (-10) = 34 8.) Find B-1  EMBED Equation.DSMT4  9.) Find A + B  EMBED Equation.DSMT4  10.) Find A ( B  EMBED Equation.DSMT4  11.) Find A ( B  EMBED Equation.DSMT4  12.) Can you find A ( B ?? no 13.) Although it is impossible to divide matrices, remember that  EMBED Equation.DSMT4 , because  EMBED Equation.DSMT4  is the multiplicative inverse of 2. (Division is just multiplication by an inverse!) So if we want to divide matrix A by B, we can multiply A by Bs inverse. Do that and write it in fractional form.  EMBED Equation.DSMT4  III. Multiplying by an inverse can be used to solve matrix equations. Consider the system of equations:  EMBED Equation.DSMT4  This can be represented by the matrix equation:  EMBED Equation.DSMT4  Think about solving a simple equation and compare it to solving a matrix equation.  Simple Equation: Matrix Equation: 4x = 12  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  x = 3 The solution is  EMBED Equation.DSMT4   14.) 5x + 7y = 2 15.) 57x ( 51y = 129 16.) (25.2x + 64.8y = 21.9 8x ( 2y = 9 76x + 98y = 623 24.8x ( 14.4y = 73.6  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Matrix equations are helpful in solving larger systems of equations also. Use the same method on these larger systems. Set up, solve and check matrix equations for these. Use your calculators to find the inverses.  x + y + z = 9 3x + 4y ( z = (10 7x + 5y ( 6z = (9 17.) 2x ( y + z = (9 18.) 5x ( 2y + 7z = 44 19.) 2x + 3y + 7z = 5 4x + y + 3z = 17 2x + y + 5z = 13 5x ( 7y + 9z = (6  EMBED Equation.DSMT4  (4, -5, 2)  EMBED Equation.DSMT4  No solution  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Doesnt Exist. (DEcdefg      ) * + , > ? 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FMathType 6.0 Equation MathType EFEquation.DSMT49ql5NT 5DSMT6WinAllBasicCodePagesComic Sans MSSymbolCourier NewTimes New RomanMT ExtrCompObjsuiObjInfovEquation Native  _1451798614yF`W `W a!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  1"x==3 FMathType 6.0 Equation MathType EFEquation.DSMT49q¢t5NT 5DSMT6WinAllBasicCodePagesOle CompObjxziObjInfo{Equation Native Comic Sans MSSymbolCourier NewTimes New RomanMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  1001[]"xy[]==  85   145 [] FMathType 5.0 Equation MathTy_1280757934~F`W `W Ole CompObj}iObjInfo796pe EFEquation.DSMT49q$DSMT5WinAllBasicCodePagesComic Sans MSSymbolCourier NewTimes New RomanMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  After simplifying both sidesEquation Native _1280758039F`W `W Ole CompObji    !"#$%(+,-./014789:;<=@CDEFGHILOPQRSVYZ[\]^_`cfghijknqrstuvy|}~, we get this and the identity matrix has the same effect as the 1 does. 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