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# MATH 1630 If high school precalculus and ACT math of at least 21 contact 694 MATH 1630

pellissippi state community college

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MATH 1630 Chapter 1 Solving Systems of Linear Equations To solve a linear system of two equations means to find all ordered pairs of real numbers which satisfy both equations Using Substitution Solve 742X7y x 7 2y 2 a Solve one of the equations for one of the variables x 2y 2 Choose the equation which is easier to solve b Substitute this quantity for x in the other equation 4 22y 2 y c Solve for y 4 4y 4 y 8 3y y 3 d Use back substitution to find x x 2 gj 2 ig 2 7 e Check your results by substituting into both original equations 42i2ii2j 1214 J 20 8 12 1O 16 6 4 74 7 2 2 3 3 3 3 gt 3 f The solution is 7E 3 3 Using Elimination Solve 74 zxiy x72y 2 a Write both equations in standard form 2xiy 74 x72y 2 b Use multiplication to create opposite M 2 7 4X 2y 8 coefficients on one of the variables X 7 2y 7 2 Solving Systems of Linear Equations p 1 J Ahrens 2006 c Add like terms and solve for x 41x 2y 8 x72y 73x 10 Xn 3 it u 10 d Backsubstltuteto nd y r72y2 10 72 2 V 3 zy E 72 y 3 e Check as above f A system with a single solution is consistent and independent Graphically 1m Hot mt x 13 42X391 2 2 E 2H II 392 3 Intersection 9333333 2EEEEE 10 10 10 10 393 333333333 392 66666666 Fins Fr39ac Flns gt Free 163 83 c IO Occasionally the calculator cannot convert a rational answer to a decimal due to a round off error Convert answers to 39a t39 ns NOTE Solving Systems of LinearEquations p 2 J A en52006 1 Using Augmented Matrices to Solve Systems of Linear Equations Elementary Row Operations x 5y 7 2 711 32 12 2x 4y 7 22 8 To solve the linear system algebraically these steps could be used All of the following operations yield a system which is equivalentto the original Equivalent systems have the same solution x 5y 7 2 711 2x 4y 7 22 8 32 12 Interchange equations 2 and 3 x5y72711 00 Multiply equation 3 by 2x 4y 7 22 21 x 5y 7 2 711 Multiply equation 2 by 7 7x 7 2y 2 74 21 lx5y72711 3y715 24 Add equation 1 to 2 and replace equation 2 with the result x5y72711 Multiply equation 2 by g y 75 2 4 x 7 2 14 Multiply equation 2 by 75 and add it y 75 to equation 1 replace equation 1 with Z 7 4 the result 7 x 18 Add equation 3 to equation 1 replace y 75 equation 1 with the result Z 7 4 The solution is 18 75 4 Augmented Matrices page 1 l 9 P Uquot 9 Operations that Produce Equivalent Systems a Two equations are interchanged b An equation is multiplied by a nonzero constant c A constant multiple of one equation is added to another equation Matrices Amatrix is a rectangular array of numbers written within brackets The size of a matrix is always given in terms of its number ofrows and number of columns in that order A 2 x 4 matrix has 2 rows and 4 columns Square matrices have the same number of rows and columns A matrix with a single column is called a column matrix and a matrix with a single row is called a row matrix A square matrix with all elements on the main diagonal equal to 1 and all other elements equal to O is called an 1 0 0 identity matrix The 3x3 identity matrix is 0 1 0 0 0 1 The position of an element within a matrix is given by the row and column in that order containing the element The element a34 is in row 3 and column 4 Elementary Row Operations that Produce RowEquivalent Matrices a Two rows are interchanged Ri e R b A row is multiplied by a nonzero constant kRi a Ri c A constant multiple of one row is added to another row kRJ Ri a Ri NOTE a means quotreplacesquot Forming an Augmented Matrix x 5y 7 z 711 32 12 2x 4y 7 22 8 An augmented matrix is associated with each linear system like 1 5 71 711 The matrix to the left of the bar is called the coefficient matrix 0 0 3 12 2 4 72 8 Solving an Augmented Matrix To solve a system using an augmented matrix we must use elementary row operations to change the coefficient matrix to an identim matrix 1 5 71 711 Form the augmented matrix 0 0 3 12 2 4 72 8 1 5 71 711 Interchange rows 2 and 3 2 4 72 8 R2 9 R3 0 0 3 12 Augmented Matrices page 2 MATH 1630 Chapter 1 Solving Systems of Linear Equations To solve a linear system of two equations means to find all ordered pairs of real numbers which satisfy both equations Using Substitution Solve 742X7y x 7 2y 2 a Solve one of the equations for one of the variables x 2y 2 Choose the equation which is easier to solve b Substitute this quantity for x in the other equation 4 22y 2 y c Solve for y 4 4y 4 y 8 3y y 3 d Use back substitution to find x x 2 gj 2 ig 2 7 e Check your results by substituting into both original equations 42i2ii2j 1214 J 20 8 12 1O 16 6 4 74 7 2 2 3 3 3 3 gt 3 f The solution is 7E 3 3 Using Elimination Solve 74 zxiy x72y 2 a Write both equations in standard form 2xiy 74 x72y 2 b Use multiplication to create opposite M 2 7 4X 2y 8 coefficients on one of the variables X 7 2y 7 2 Solving Systems of Linear Equations p 1 J Ahrens 2006 c Add like terms and solve for x 41x 2y 8 x72y 73x 10 Xn 3 it u 10 d Backsubstltuteto nd y r72y2 10 72 2 V 3 zy E 72 y 3 e Check as above f A system with a single solution is consistent and independent Graphically 1m Hot mt x 13 42X391 2 2 E 2H II 392 3 Intersection 9333333 2EEEEE 10 10 10 10 393 333333333 392 66666666 Fins Fr39ac Flns gt Free 163 83 c IO Occasionally the calculator cannot convert a rational answer to a decimal due to a round off error Convert answers to 39a t39 ns NOTE Solving Systems of LinearEquations p 2 J A en52006 Matrix Arithmetic on the Tl83 The size dimension ofa matrix is always given in terms of its number of rows and number of columns A 2 X 4 matrix has 2 rows and 4 columns Square matrices have the same number of rows and columns Only matrices ofthe same size can be added or subtracted There are an in nite number of special square matrices called identity matrices An identity matrix I has 15 on the main diagonal running from upper left to lower right and Os elsewhere Some square matrices have inverses lfA 1 and A are inverse matrices then A 1A A A 1 equals the identity matrix I ofthe appropriate size Any matrix may be multiplied by any real number called a scalar Each element ofthe matrix is multiplied by the scalar Two matrices can be multiplied ifand only ifthe number of columns in the first matrix is the same as the number of rows in the second Any square matrix can be raised to a power A zero matrix 0 may be ofany size and has Os as all of its elements Entering a Matrix To enter a 2 x3 matrix as matrix A NAMES 1 A MATR l XA 2 LENIERJ 3 LENIERJ LENIERJ LENIERJ 0 LENIERJ Displaying a Matrix MINAMES 1A m1 m1 Adding or Subtracting Matrices NAMES 1A NAMES 2B Multiplying a Matrix by a Scalar NAMES 1 A Multiplying Matrices NAMES 1A NAMES 2 3 Raising a Matrix to a Power MNAMES 2 BJWEW Matrix Arithmetic page 1 Entering an Identity Matrix One way to enter an identity matrix is to simply enter the required elements as you would any other 39 A matrix Ashortcut for entering the 3x 3 identity matrix Is M TH 5 identity m MATRIX NAMES 4 D ENTER Examples of Matrix Calculations Enter these matrices A g A B B A Addition of matrices is commutative ie A B B A B B A Subtraction of matrices is not commutative ie A B o B A A F does not exist because the matrices do not have the same size 2A A 2 Scalar multiplication ofa matrix is commutative ie kA Ak 5BE 5B5E Scalar multiplication is distributive over matrix addition ie kB E kB kE Scalar multiplication is also distributive over matrix subtraction CI IC The product ofa square matrix and the identity matrix ofthe same order is the original matrix Such multiplication is commutative ie CI 10 AB BA Multiplication of matrices is not commutative ie AB BA BE The product of two matrices may be 0 even if neither matrix is 0 FC CF does not exist because the number of columns in C is not the same as the number of rows in F The size ofthe product of two matrices assuming the product exists is the number of rows in the rst matrix by the number of columns in the second A2 AA B 2 The square ofa nonzero matrix may be 0 G 1 convert to fractions by using MATH 1 Matrix Arithmetic page 2 MATRIX ARITHMETIC ON THE TI83 Given the following matrices A j B and C 2 5 1 You will enter matrices A B and C in your calculator as follows A MATRIX ENTER Arrow over to EDIT ENTER B C Enter the size as 2 X 2 this is rows by columns ENTER D Type over the values already there be sure to hit enter each time E 2 QUIT to get back to the home screen F Repeat for matrix B except move down to B on EDIT before hitting enter G Repeat for matrix C except move down to C on EDIT before hitting enter H 2 QUIT to get back to the home screen MFITRIXIFI 2 x2 MFITRIXIB 2 x2 W E E1 h I E a h g I 2 1 1 22B 221 25 2 Toaddtwo matricesAB mHB 2 3 3 1 In the home screen MATRIX A ENTER 1 MATRIX B ENTER 3 To multiply a matrix by a scalar 2B 3A 2B 3FI 6 1 11 2 In the home screen 2 MATRIX B ENTER 1 3 MATRIX A ENTER 4 To multiply matrices C H 391 2 A CA 7 MATRIX C ENTER MATRIX A ENTER B Ac 7 MATRIX A ENTER MATRIX CENTER What went wrong here HANDOUT MATRIX MULTIPLICATION The scalar or dot product and its properties Given the ntuples u 1 3 1 v 1 4 2 and w231 The scalar or dot product u v is defined to be 11 34 12 9 v u1143219 so uvv u kuvk131vk3k1k142k12k2k9k kuvk99k ukvuk142131k4k2kk12k2k9k sokuvkuvukv vw142231113and vw1311131331 uv9anduw1312312918souvuw981 Thusuvwuvuw A word of caution We have only shown these properties to be true for one specific set ofordered triples All of these properties can be shown to be true see Handout What does the dot product of ntuples have to do with matrix multiplication I m glad you asked because it turns out that the dot product is the very mechanism used to multiply matrices 2 1 0 1 1 Suppose that we are asked to find C AB where A 1 1 1 and B 0 1 Since these 71 4 3 71 3 matrices have different dimensions you probably would think they can t be multiplied We will address that issue in a few minutes For now assume they can be multiplied 1 1 Let U 2 1 O and V O Then UV 2 1 O O and VT 1 O 1 The definition for 71 71 multiplying a row matrix times a column matrix says that the product is actually the dot product UVT u v 210101 2 O O 2 This process can actually be streamlined quite a bit as follows 9 Ag A 2110012 1 Remember RC The product of row 1 of the first matrix and column 1 of the second gives us element C11 in the product C AB To complete the product AB we must find 1 1 c11210 O 21 10 O1 2 c12 210 1 21 11 O3 3 3 1 1 cz1111 o 11101 1o c22 111 1 1111135 71 3 1 1 0311 43 o 11403 1 4 c32 1 4 3 1 11413312 3 MATH 2000 Matrix Multiplication page 1 J Ahrens 1192005 2 3 Therefore C AB 0 5 74 12 What are the characteristics of matrix multiplication Let s return to a previous concern Why is multiplication possible in this case Is there a general rule thatwill predict when the multiplication of two matrices is possible and when it is not Is there a rule which will predict the dimensions of a matrix product assuming it exists Since each element in the product is the result of multiplying a row from the first matrix times a column from the second it follows that the number of elements in each row of the first matrix must equal the number of elements in each column of the second matrix This is equivalent to saying that the number of columns in the first matrix must equal the number of rows in the second matrix OK so RC didn t work in this case Looking at the dimensions of the matrices is very helpful A B 3 2 Ifthese two inner numbers match then the product AB exists Do the outer numbers have any significance As a matter of fact they predict the size of of the product A B 3x3 3x2 f The product AB will be a 3x2 matrix Why doesn t BA exist B A 3 2 BA doesn t exist because the number of columns in the first matrix does not equal the number of rows in the second ls matrix multiplication commutative Obviously not since it may not even be possible to multiply in both orders Perhaps matrix multiplication is commutative if both products exist and B 73 then 3 4 o 75 AB 122o 1732752 713 and 3240 373475 6 729 21733 227 34 7 78 BA 01753 02754 715 720 OK so matrix multiplication is not commutative in general We will see later that there is a very special case where matrix multiplication will be commutative Stay tuned MATH 2000 Matrix Multiplication page 2 J Ahrens 1192005 Several familiar properties of multiplication of real numbers do apply to matrix multiplication Matrix multiplication is associative ie if ABC is defined then ABC ABC lf AB is defined then kAB kAB AkB the product of a scalar and a matrix product is commutative If all indicated operations can be performed then AB C AB AC and A B C AB AC ie matrix multiplication is distributive over matrix addition or matrix subtraction Don t change the order of matrices Prove that if all indicated operations can be performed then AB C AB AC Proof for the case when A is 3x2 and B and C are 2x2 a a 11 12 b b c c GivenA a21 2122B11 12 andC 11 12 b21 22 c21 c22 331 332 Prove that AB C AB AC B c b114rc11 b12C12 b21 c21 b22 t 022 definition of matrix addition ABC a a 11 12 b11c11 321 322 b12 012 b214r c21 b22 022 331 8111b114r 011 t 8112b214r c21 311b12 012 t 312b22 022 8121b114r 011 t 8122b214r c21 321b12 012 t 322b22 022 8131b114r 011 332b214r c21 a31b12 012 a32b22 022 definition of matrix multiplication 5 111b114r 3110114r 5 112b214r 312021 a11b12 311012 a12b22 a12 22 321b11 81210114r 322b21 322021 321b12 321012 322b22 322022 331b11 81310114r 332b21 332021 331b12 331012 332b22 332022 distributive property of multiplication over addition for real numbers a11b114r a12b21 a11b12 a12b22 AB a21b11 a22b21 a21b12 a22b22 defInItIon of multIplIcatIon of matrIces 331b11 332b21 331b12 332b22 311 12a12 21 311 12a12 22 AC a21c11 a22c21 a21c12 a22c22 definition of multiplication of matrices 81310114r 232 21 331012 a32 22 a11b114r a12b214r 311012 312021 a11b12 a12b22 311012 312022 AB AC 321b11 322b21 81210114r 322021 321b12 322b22 321012 322022 331b11 332b21 81310114r 332021 331b12 332b22 331012 332022 definition of addition of matrices MATH 2000 Matrix Multiplication page 3 J Ahrens 1192005 a11b114r 81110114r a12b214r 312021 a11b12 311012 a12b22 312022 321b11 81210114r 322b21 322021 321b12 321012 322b22 322022 331b11 81310114r 332b21 332021 331b12 331012 332b22 332022 a11b114r a12b214r 311012 312021 a11b12 a12b22 311012 312022 321b11 322b21 81210114r 322021 321b12 322b22 321012 322022 331b11 332b21 81310114r 332021 331b12 332b22 331012 332022 associative property of addition of real numbers Therefore AB C AB AC underthe limitations stated at the beginning Some examples of matrix multiplication 72 1 2 72 fD 8 3 7 andE 1 0 then 6 74 5 71 2 4 72211074 72721005 3 4 DE 82731774 87273075 15 19 025194 025095 42 745 71221474 71722045 16 22 ED does not exist why not IfF12G2 3 andl10 34 075 01 G 7 and IG 7 lwill be discussed more later in this section 715 720 Hmmm even if both matrix products exist they may not be equal ie matrix multiplication is not commutative fH13 andK 171thenHK 7 7 butKH 0 0 13 2 72 7 77 o 0 Wait Something is not fair here How can the product of two matrices be 0 if neither matrix is 0 Because matrices do not play by the same rules as real numbers So much for the oft used Zero Product Property FG 2 713butGF 7 8 6 729 So what s a poor student to do Which of the familiar properties of real number arithmetic are also true for matrix arithmetic In general you can change the grouping of matrices in an operation but not the order The proof of the validity ofa matrix operation is frequently based on the properties of real numbers lfyou can find even one example where a given property fails to be true then that is not a valid property for matrices We have already found some examples where matrix multiplication failed to be commutative Here is a glimpse of why multiplication of 2 x 2 matrices is not commutative MATH 2000 Matrix Multiplication page 4 J Ahrens 1192005 IfAa11 a12 andBb11 bu 2121 2122 b21 b22 PABT11 am bm b12a11b11a12b21 a11b12a12b22 and 2121 2122 b21 b22 a21b11 a22b21 321b12 a22b22 Q BAb11 b12Ma11 2112 b11311 b12321 b21 b22 a21 b11312 b12322 a22 b21311 b22321 b21312 b22322 definition of multiplication of matrices By the definition of equality of matrices each pair of corresponding elements must be equal In particular we must have p11 q11 3 a11b11 a12b21 b11811 b22821 Commutativity of real numbers guarantees that a11b11 b11a11 but there is no way to guarantee that a12b21 b22a21 Therefore commutativity of multiplication of 2 x 2 matrices fails More matrices with special properties The multiplicative identity matrix I has the following properties is a square matrix All elements on the main diagonal are 1s and all other elements are Os IB B and BI B provided those products exist IB Bl B if B is a square matrix f 1 0 andX 1 72 0 thenX 1 72 0 butdeoesnotexist o 1 4 8 71 4 s 1 fY 175thenlY 1 5 andY 1 5 9 o 9 o 9 o The zero matrix 0 additive identity has the following properties A 0 0 A A provided those sums exist A0 0A 0 provided those products exist Remember that AB 0 does not necessarily imply that either A or B 0 Triangular matrices must be square ln lower triangular matrices nonzero elements can appear only on or below the main 1 0 0 diagonal eg 2 0 0 74 0 6 ln upper triangular matrices nonzero elements can appear only on or above the main 0 O 1 diagonaeg O O O O O O MATH 2000 Matrix Multiplication page 5 J Ahrens 1192005 Matrix Arithmetic on the Tl83 The size dimension ofa matrix is always given in terms of its number of rows and number of columns A 2 X 4 matrix has 2 rows and 4 columns Square matrices have the same number of rows and columns Only matrices ofthe same size can be added or subtracted There are an in nite number of special square matrices called identity matrices An identity matrix I has 15 on the main diagonal running from upper left to lower right and Os elsewhere Some square matrices have inverses lfA 1 and A are inverse matrices then A 1A A A 1 equals the identity matrix I ofthe appropriate size Any matrix may be multiplied by any real number called a scalar Each element ofthe matrix is multiplied by the scalar Two matrices can be multiplied ifand only ifthe number of columns in the first matrix is the same as the number of rows in the second Any square matrix can be raised to a power A zero matrix 0 may be ofany size and has Os as all of its elements Entering a Matrix To enter a 2 x3 matrix as matrix A NAMES 1 A MATR l XA 2 LENIERJ 3 LENIERJ LENIERJ LENIERJ 0 LENIERJ Displaying a Matrix MINAMES 1A m1 m1 Adding or Subtracting Matrices NAMES 1A NAMES 2B Multiplying a Matrix by a Scalar NAMES 1 A Multiplying Matrices NAMES 1A NAMES 2 3 Raising a Matrix to a Power MNAMES 2 BJWEW Matrix Arithmetic page 1 Entering an Identity Matrix One way to enter an identity matrix is to simply enter the required elements as you would any other 39 A matrix Ashortcut for entering the 3x 3 identity matrix Is M TH 5 identity m MATRIX NAMES 4 D ENTER Examples of Matrix Calculations Enter these matrices A g A B B A Addition of matrices is commutative ie A B B A B B A Subtraction of matrices is not commutative ie A B o B A A F does not exist because the matrices do not have the same size 2A A 2 Scalar multiplication ofa matrix is commutative ie kA Ak 5BE 5B5E Scalar multiplication is distributive over matrix addition ie kB E kB kE Scalar multiplication is also distributive over matrix subtraction CI IC The product ofa square matrix and the identity matrix ofthe same order is the original matrix Such multiplication is commutative ie CI 10 AB BA Multiplication of matrices is not commutative ie AB BA BE The product of two matrices may be 0 even if neither matrix is 0 FC CF does not exist because the number of columns in C is not the same as the number of rows in F The size ofthe product of two matrices assuming the product exists is the number of rows in the rst matrix by the number of columns in the second A2 AA B 2 The square ofa nonzero matrix may be 0 G 1 convert to fractions by using MATH 1 Matrix Arithmetic page 2 Row Operations and Inverse Matrices on the TI83 I Elementary Row Operations A w 0 U quot39 n To multiply row 1 of matrix A by 4 LetA 2 81 2 7 139 To interchange rows 1 and 2 of matrix A MATRIX MATH C rowaap ENTER MATRIX NAMES 1A j C E ENTER 2 7 1 The result is 2 8 1 To add rows 1 and 2 of matrix A MATRIX MATH D row ENTER MATRIX NAMES 1A C C I ENTER 2 8 1 O 1 0 operation replaces row 2 The result is Row1 remains unchanged while the result ofthe row MATRIX MATH Erow C MATRIX NAMES 1A j I ENTER 8 32 4 The result Is 2 7 1 To multiply row 1 of matrix A by 5 and add it to row 2 MATRIX MATH F row ENTER C MATRIX NAMES 1A C I ENTER 2 8 1 The result Is 2 33 4 operation replaces row 2 Row1 remains unchanged while the result ofthe row The result of row operations is displayed on the screen but it is not stored It is usually desirable to store the result for further operations To store the result from the lastoperation performed as F MATRIX NAMES 6F ENTER ll Inverse Matrices A B Ifa square matrix has an inverse it is said to be invertible nonsingular If A 1 and A are inverse matrices then A A 1 A 1A I 5 1 6 26 LetA 1 1 4andB 1 5 22 2 O O O 1 1 O O 53 52 26 Since AB 0 1 0 and BA 44 43 22 A and B are not inverses O 2 8 2 2 O 3 7 9 73 C LetA 12 27 andB 4 1 Since AB a and BA a A and B are inverse matrices Ill Creating an Identity Matrix A An identity matrix I can be created by entering the proper elements as with any other matrix B The shortcut for creating the 3X3 identity matrix and storing it as matrix A is MATH 5 identity 3 NAMES 1A IV Augmenting a Matrix with Another Matrix 0 1 1 7 l o 1 1 7 l A LetA 2 2 1 andB 0 Theaugmented matrix 2 2 1 o 0 O O 1 3 O 1 o 3 can be created and stored as matrix C without altering the original matrices as follows MATH 7augment NAMES 1A D NAMES 2B I NAMES 1 C B It is frequently useful to create an augmented matrix using an identity matrix The O 1 1 1 O O O 1 1 augmented matrix 2 2 10 1 0 can be formed by lettingA 2 2 1 O 0 10 0 1 O O 1 1 O O andB 0 1 0 and usingthe methodofthe previousexample O O C Ashortcut for augmentingasquare matrix with an identity matrix MATRIX MATH 7au ment MATRIX NAMES 1A j MATRIX MATH 5identity amp ENTER MATRIX NAMES 3C ENTER V GaussJordan Elimination A Perform any row operation which will yield element a 11 1 Avoid creating fractions whenever possible B Use the rst row and row operations to convert all other elements in column 1 to zeros C Multiply row 2 by a scalar to yield element a 22 1 Sometimes there are other options available but be careful that you do not change any elements in column 1 D Use row 2 and row operations to covert all other elements in column 2 to zeros E Repeat this process until all columns have been converted VI Finding an Inverse Matrix Using an Augmented Matrix and Elementary Row Operations A B If a square matrix A has an inverse it can be found by augmenting A with the appropriately sized identity matrix I and then using the GaussJordan elimination method to transform the matrix to the left ofthe bar into the identity matrix I Ifthis cannot be done then A is singular and has no inverse fA OON OO ONO CAN 1 1 O 1 1 0 1 1 1 0 0 2 1 then the required augmented matrix is 2 2 1 o O 1 O 0 1 0 0 10 0 1 1 0 1 0 R1 R2 1 1 0 0 10 0 1 120 12 0 R1 R1 1 1 0 0 1 0 0 1 The rest ofthe elements in column 1 already equal zero 2 Element a 22 already equals 1 1 0 12 1 12 0 1R2 R1 R1 0 1 1 0 0 0 0 0 0 1 Column 2 is now completed 3 Element a 33 already equals 1 1 o o 1 12 12 R3 R1 R1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 12 12 0 1 o 1 o 1 R3 R2 R2 0 0 0 0 1 Column 3 is now completed 4 This means that A 1 1 O O 1 1 12 12 0 1 5 To prove the result is correct show that AA 1 1A 004 040 OO I VII Shortcuts for Finding an Inverse Matrix A To nd the row reduced form ofthe augmented matrix A from above MATRIX MATH B rref MATRIX NAMES 1A I ENTER B To nd A 1 directly begin with A not augmented MATRIX NAMES 1A ENTER 3 7 1 9 73 C lfA 12 27 thenA 4 1 3 6 1 D lfC 2 4 thenC doesnoteXIst 1 O O 1 O O E le 010thenE 1 010 O O 1 O O 1 VIII Determinants A Although only square matrices can have inverses there are some square matrices which do not have inverses Each square matrix has associated with it a real number called its determinant and the determinant determines whether or not an inverse matrix exists w The determinant ofthe 2X2 matrix ad bc a C is denoted by 2 g and is equal to b d 3 1 lfA 12 27 then A 3 27 12 7 3 2 lfB12 5 5 10 thenB12 10 55 95 3 6 3 lfC 2 4 then C 34 2 6 o C Although it is possible to find determinants of larger matrices by hand it can be tediousandtimeconsuming lfA 1g 27 the calculatorcan be used to nd A MATRIX MATH 1det MATRIX NAMES 1A I ENTER 1 0 0 D 50 1 0 51 0 0 1 E Determinants may be used to write inverse matrices 3 7 1 39 12 27393quotquot1ndA 39 1 277 3 123 Awlw 9 4 m The denominator ofthe scalar is the determinant ofthe original matrix b This representation usually eliminates fractions from the inverse matrix 0 Since the determinant is in the denominator of the scalar any matrix with a zero determinant will fail to have an inverse Such a matrix is singular 2 5 1 2 5 1 2 Since 5 2 7 1 52 so 2 7 1 isinvertible 1 1 1 1 1 1 4 3 1 2 5 1391 1 1 0 1 8 6 2 2 7 1 2 2 1 1 o 1 1 1 2 1 2 2 9 7 4 2 2 IX Using Matrix lnverses to Solve Equations A A linear system in standard form can be expressed as the matrix equation AX B where A is the matrix of coef cients ofthe variables X is the column matrix ofthe variables and B is the column matrix of constants appearing to the right of the equal signs B lfA is nonsingular then the system s solution is given by X A 1B The inverse matrix A 1 must be placed to the left of B as shown 125y3 C To solve the system 5X 2y 4 usIng matrIces 1 12 5 x 3 o x 12 5 3 5 2 y39andB39 4 y 5 2 4 2 This means the solution is X 26 and y 63 Check by substituting these values into both equations 1A X 6339 461 D lfthe coefficient matrix A is singular then the system has no solution 2X 8yz 5 2 8 1 X 5 E For 2x7y z 3A 2 7 1X andB 3 x yz 1 1 1 1 y 1 2 8 1 1 5 8 9 1 5 14 1XA 1B 2 7 1 3 1 1 o 3 2 1 1 1 1 9 1o 2 1 17 2 This means that X 14 y 2 and 2 17 X Cramer s Method A Another method for solving systems of simultaneous equations uses determinants The system must be square Number of equations number ofvariables 2X 8y z 5 B For 2x7y z 3 x y z 1 5 8 1 5 3 7 1 5 1 X 1 11 2 8 1 1 5 2 7 1 g 1 1 1 The determinant on the bottom is the coef cient determinant The determinant on top is the same as the coef cient determinant except for the X column which is the column to the right of the equal signs 2 5 1 5 2 3 1g 2y111i2 8 1 1

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