# 1.4 - Dr Doom Lin Alg MATH185O Section 1.4 Page 1 of 7...

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Unformatted text preview: Dr. Doom Lin Alg MATH185O Section 1.4 Page 1 of 7 Chapter 1 - Linear Equations and Matrices Section 1.4 - Inverses; Rules of Matrix Arithmetic We look at some properties of arithmetic operations on matrices. While many of the rules of arithmetic for real numbers hold for matrices as well, some do not. Matrix multiplication is not commutative! l.e. AB and BA need not be equal. Examples: 2—1 0 1 (1)W1thA=[ 0 0 _2 iandB=i”‘11 1:1. 1-1-3 O o '1 2‘3 2&1. A@ = [1 q a} is HOT berm» At :[1] %A=[1 lx't— 1M 2M lat-1— mm MU . . 10 (1V)W1thA—|:0 1],andB— 0 5] 1 0 l (iii)WithA=[0 0],andB=H 8] 1 Avg-«[1 ° =&A Dr. Doom Lin Alg MATH185O Section 1.4 Page 2 of 7 Theorem 1.4.1 Properties of Matrix Arithmetic Assuming that the sizes of the matrices are such that the operations below can be performed, the following rules hold: (a) A —— B = B + A (Commutative law for addition) (b) A —— (B + C’) = (A + B) + C' (Associative law for addition) (c) A(BC’) = (AB)C' (Associative law for multiplication) (d) A(B + C') 2 AB + AC’ (Left distributive law) (e) (A + B)C’ 2 AC’ + BC’ (Right distributive law) (f)A(B—O)=AB—AO (j) (a+b)C'=aC'+bC’ (g)(B—C')A=BA—C'A k)(a—b)C’=aC'—bC’ (h) a(B + C') 2 (LB + a0 (l) a(bC') = (ab)C' (i) a(B — C') 2 (LB — aC’ a(BC') = (aB)C' = B(aC') @[email protected] 4% A P m 6— 0 g/ v :5 II p A 6— Q U 5? A S" (3 «1". II ‘p ‘ r g/ (x :5: ll 45? 5 a v :5: = (a. \Oc1)C,_\-) + .. . +(Qioc¥rs r. (akyh C45 l— (QB)C1C13+"'+ (Q‘slzrcr‘s [email protected] Dr. Doom Lin Alg MATH185O Section 1.4 Page 3 of 7 Deﬁnition A matrix whose entries are all equal to zero is called a zero ma- triX. Examples: 0 _ : O =[ I 2x; 000 The cancelation law does not hold for matrices. That is AB 2 AC DOES NOT imply that B = C', and AD 2 O DOES NOT imply that A = O or D = 0. Example: A=[8 31],B=[3 A9,=[‘3 *‘\ AC=[‘3 "K 50 AgcAC gar we. 4 L Theorem 1.4.2 Properties of Zero Matrices Assuming the sizes of the matrices are such that the operations below can be performed, the following rules are valid: 6M A is mum Dr. Doom Lin Alg MATH185O Section 1.4 Page 4 of 7 Deﬁnition A square matrix with 1’s on the main diagonal and 0’s every- where else is called an identity matrix, and is denoted by I, or by In when we need to emphasize that the size of the identity is n X n. Examples: Theorem 1.4.3 If R is the reduced row-echelon form of an n X n matrix A, then either R has a row of zeros or R is the identity matrix In. Q. HRS en'th tXAc‘TLV IA Lew-u cues, Tau—:1.) ‘15:“ 02. \* HA6 <vx Least-Sc ones mum) rm; Wu) THKT HMS pa new“, A conscien- cynic!» a1- 7.5%. Deﬁnition If A is a square matrix, and if a matrix B of the same size can be found such that AB 2 BA 2 I, then A is said to be invertible and B is called an inverse of A. If no such B exists, then A is said to be singular. Examples: . . 2 3 —1 —3 (1)W1thA—[_1_1],andB—[ 1 2] Ag_[ l 01:: = 3A so A is Au \Suemeu: m (So is k) 0 l .. . 1 0 a b (11)W1thA—|:0 0],andB—[C d] ABC 75 :1 7%0‘1‘0194' so A F". ALL CAD Dr. Doom Lin Alg MATH185O Section 1.4 Page 5 of 7 Theorem 1.4.4 If B and C’ are both inverses of the matrix A, then B = C'. Ac> —. Ma, AC =CA=1 g)C Mg 05050565 at- Anew; 5: 19, [email protected])t=c(AL3= (.1: c If the matrix A is invertible, we denote its inverse by 14—1. Theorem 1.4.5 The matrix a b A - l c d l is invertible if and only if ad — be 7é 0, in which case _ 1 d —b _ad—bc —c a ' Theorem 1.4.6 If A and B are invertible matrices of the same size, then AB is invertible, and A—l (AB)—1 = B—lA—l. LAeXr‘A“) = Mw‘m" -. A1 A" = AA" ,1- M cuecv. Lg"A"XAe,)= X 661\$qu Tull. Tun To MW 99-0” 9" mwmw‘e mugs (A\A»--- AA? A;‘-.-A;‘A:' WHGLE AHA”... ,A“ At: ilweﬁisa Was a: WE sC‘LE. hL/X: \$0ng Dr. Doom Lin Alg h-‘lATHlEi-BU Section 1.4 Page 6 of 7 Example: 2 3 3 —1 21 —8 C "'l‘Az .3: .AB= OIJbIC e1 [1 2] [5 _. I [13 _5] 1-1- 34 = \fO So A15 misenhxew . -\ A = 1-7. ‘3'\ (5’: \ [—1 l -\ t [ 7. ’l‘k @‘A‘\— S ,3} a 177') “('95 “S 2“ S — J \‘b -'2\ g ,\ \ —g g :— 42 :- -'\ —\ FOL -; __’- [—13 1:\ [Vs dig @ 2\C-§\- (-013 Deﬁnition Powers of a Matrix If A is a square matrix then we deﬁne the non—negative integer powers of A to be: n factors Moreover, if A is invertible, then we deﬁne the negative powers of A to be A0 = I, A” (n > U). A‘" = (A‘1)". Theorem 1.4.7. Laws of Exponents If A is a square matrix and 7’ and s are integers, then ATAS = AT+S7 (A7")S 2 ATS. Theorem 1.4.8. Laws of Exponents If A is an invertible matrix, then (a) A‘1 is invertible and (A‘1)_1 = A; (b) A" is invertible and (An)—1 = (A4)” for n = 0, 1, 2, . . .; (C) For any non-0 scalar k, the matrix [CA is invertible, and (kA)‘1 = %A_1. S»! lL'fO scout, A Is lilvewnlsLE @A)(‘—k_A")=(k't)(AK‘)=l-1 =1 Ace» cm G;A"XLA§=1. Dr. Doom Lin Alg MATH185O Section 1.4 Page 7 of 7 Polynomial Expressions Involving Matrices _ 2— _ 3 —1 p(a:)—a: 333+1, A—[O 2] = XZ-i’ax + X° p(A\-— AI’BA+£ I Theorem 1.4.9. Properties of the Transpose. If the sizes of the matrices are such that the operations can be performed, then (a) (AT T = A; (b) (A :: B)T 2 AT :: BT; (c) (kA)T = MAT), Where k is any scalar; (d) (AB)T = BTAT. In fact the transpose of an arbitrary product of matrices is the product of the transposes, in reverse order.) Q (0931)“ t OLA)" = MA)“ = Lamas t (Emit . ® (@9353 = (14% = @5301; +Q5Llola+ . -_ + 43AM, (Assume 1m éu mm) is rm = ‘9“. “Hi + \azaqst 1' ' "‘\' \o'iqsr = (Gar Ages , ’J'ia' Theorem 1.4.10. If A is an invertible matrix, then so is AT, and (ATP = W)? LAWCK‘Yt (K‘A3T : :[T 1 ms: cued. @"\TAT=:[. ll ...
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## This note was uploaded on 11/10/2009 for the course MATH 1850 taught by Professor Mihaibeligan during the Spring '09 term at UOIT.

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1.4 - Dr Doom Lin Alg MATH185O Section 1.4 Page 1 of 7...

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