JUST THE MATHS SLIDES NUMBER 9.3. MATRICES 3 (Matrix inversion & simultaneous equations) A.J.Hobson

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1 JUST THE MATHS SLIDES NUMBER 93 MATRICES 3 (Matrix inversion & simultaneous equations) by AJHobson 93 Introduction 932 Matrix representation of simultaneous linear equations 933 The definition of a multiplicative inverse 934 The formula for a multiplicative inverse

2 UNIT 93 - MATRICES 3 MATRIX INVERSION AND SIMULTANEOUS EQUATIONS 93 INTRODUCTION In Matrix Algebra, there is no such thing as division in the usual sense An equivalent operation called inversion is similar to the process where division by a value, a, is the same as multiplication by a For example, consider the equation The solution is obviously Alternatively, mx = k x = k m (a) Pre-multiply both sides of the given equation by m (b) Rearrange this as m (mx) = m k (m m)x = m k

3 (c) Use m m = to give (d) Use x = x to give x = m k x = m k Later, we see an almost identical sequence of steps, with matrices Matrix inversion is developed from the rules for matrix multiplication 932 MATRIX REPRESENTATION OF SIMULTANEOUS LINEAR EQUATIONS In this section, we consider three simultaneous linear equations in three unknowns a x + b y + c z = k, a 2 x + b 2 y + c 2 z = k 2, a 3 x + b 3 y + c 3 z = k 3 2

4 These can be written as or a b c a 2 b 2 c 2 a 3 b 3 c 3 x y z = k k 2 k 3 MX = K Note: Suppose N is such that NM = I Pre-multiply MX = K by N to give That is, N(MX) = NK (NM)X = NK In other words, Hence, IX = NK X = NK N exhibits a similar behaviour to the number m encountered earlier; we replace N with M 3

5 933 THE DEFINITION OF A MULTIPLICATIVE INVERSE The multiplicative inverse of a square matrix M is another matrix, denoted by M which has the property M M = I Notes: (i) It is certainly possible for the product of two matrices to be an identity matrix (see Unit 92, Exercises) (ii) We may usually call M the inverse of M rather than the multiplicative inverse (iii) It can be shown that, when M M = I, it is also true that MM = I (iv) A square matrix cannot have more than one inverse Assume that A had two inverses, B and C Then, C = CI = C(AB) = (CA)B = IB = B 4

6 934 THE FORMULA FOR A MULTIPLICATIVE INVERSE (a) The inverse of a 2 x 2 matrix Taking M = a b a 2 b 2 and M = P R Q S, we require that a b a 2 b 2 P R Q S = 0 0 Hence, a P + b R =, a 2 P + b 2 R = 0, a Q + b S = 0, a 2 Q + b 2 S = 5

7 These equations are satisfied by P = b 2 M, Q = b M, R = a 2 M S = a M, where M = a b a 2 b 2 M is called the determinant of the matrix M Summary M = M b 2 b a 2 a 6

8 EXAMPLES Write down the inverse of the matrix 5 3 M = 2 7 Solution Hence, Check M M = 4 M = 4 M = = =

9 2 Use matrices to solve the simultaneous linear equations 3x + y =, x 2y = 5 Solution The equations can be written MX = K, where M = 3, X = 2 First, check that M 0 Thus, M = x y and K = 3 = 6 = 7 2 M = The solution of the simultaneous equations is given by x y That is, = = 5 7 x = y = = 2 8

10 (b) The inverse of a 3 x 3 Matrix We use another version of Cramer s rule The simultaneous linear equations a x + b y + c z = k, a 2 x + b 2 y + c 2 z = k 2, a 3 x + b 3 y + c 3 z = k 3 have the solution x k b c k 2 b 2 c 2 k 3 b 3 c 3 = y a k c a 2 k 2 c 2 a 3 k 3 c 3 = z a b k a 2 b 2 k 2 a 3 b 3 k 3 = a b c a 2 b 2 c 2 a 3 b 3 c 3 9

11 METHOD (i) The last determinant above is called the determinant of the matrix M, denoted by M In M, we let A, A 2, A 3, B, B 2, B 3, C, C 2 and C 3 denote the cofactors (or signed minors ) of a, a 2, a 3, b, b 2, b 3, c, c 2 and c 3 respectively (ii) For each of k, k 2 and k 3 the cofactor is the same as the corresponding cofactor in M (iii) The solutions for x, y and z can be written as follows: x = M (k A + k 2 A 2 + k 3 A 3 ) ; y = M (k B + k 2 B 2 + k 3 B 3 ) ; z = M (k C + k 2 C 2 + k 3 C 3 ) ; or, in matrix format, x y = M z Compare this with A A 2 A 3 B B 2 B 3 C C 2 C 3 X = M K 0 k k 2 k 3

12 We conclude that M = M A A 2 A 3 B B 2 B 3 C C 2 C 3 Summary Similar working would occur for larger or smaller systems of equations In general, the inverse of a square matrix is the transpose of the matrix of cofactors times the reciprocal of the determinant of the matrix Notes: (i) If M = 0, then the matrix M does not have an inverse and is said to be singular If M 0, M is said to be non-singular (ii) The transpose of the matrix of cofactors is called the adjoint of M, denoted by AdjM There is always an adjoint though not always an inverse When the inverse exists, M = M AdjM

13 (iii) The inverse of a matrix of order 2 2 fits the above scheme also The cofactor of each element will be a single number associated with a place-sign according to the following pattern: + + Hence, if then, M = M = a b a 2 b 2 a b 2 a 2 b, b 2 b a 2 a The matrix part of the result can be obtained by interchanging the diagonal elements of M and reversing the signs of the other two elements 2

14 EXAMPLE Use matrices to solve the simultaneous linear equations 3x + y z =, x 2y + z = 0, 2x + 2y + z = 3 Solution The equations can be written MX = K, where M = X = x y z and K = 0 3 M = = 3( 2 2) ( 2)+( )(2+4) M = 7 3

15 If C denotes the matrix of cofactors, then C = Notes: (i) The framed elements indicate those for which the place sign is positive (ii) The remaining four elements are those for which the place sign is negative (iii) In finding the elements of C, do not multiply the cofactors of the elements in M by the elements themselves The Inverse is given by M = M AdjM = 7 CT = 7 The solution of the equations is given by x y z = = = 3 5 4

JUST THE MATHS UNIT NUMBER 9.3. MATRICES 3 (Matrix inversion & simultaneous equations) A.J.Hobson

JUST THE MATHS UNIT NUMBER 9.3. MATRICES 3 (Matrix inversion & simultaneous equations) A.J.Hobson JUST THE MATHS UNIT NUMBER 93 MATRICES 3 (Matrix inversion & simultaneous equations) by AJHobson 931 Introduction 932 Matrix representation of simultaneous linear equations 933 The definition of a multiplicative