1. CHAPTER 1 Section 1.4 إعداد د. هيله العودان (وضعت باذن معدة العرض )

2. Example 3 Consider the matrices A = B=C= D= Here AB=AC= 010201021 11341134 25342534 370037000 34683468 H.A.O

3. Theorem 1.4.2. Assuming that sizes of the matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid. (a) A+ 0 = 0 +A=A (b) A-A= 0 (c) 0 -A=-A (d) A 0 = 0 ; 0 A= 0 H.A.O

4. IDENTITY MATRICES and so on. 10011001 10 0 01 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 H.A.O

5. Theorem 1.4.3. If R is the reduced row- echelon from of an n × n matrix A, then either R has a row of zeros or R is the identity matrix I n. H.A.O

6. Definition. If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, then A is said to be invertible and B is called an inverse of A. H.A.O

7. Theorem 1.4.4. If B and C are both inverses of the matrix A, then B=C. H.A.O

8. AA = I and A A= I H.A.O

9. Theorem 1.4.5. The matrix A= is invertible if ad-bc≠ 0, in witch case the inverse is given by the formula A = = abcdabcd 1 ad - bc d-b -c a d ad - bc b ad - bc c ad - bc a ad - bc H.A.O

10. Theorem 1.4.6. If A and B invertible matrices of the same size, then: (a) AB is invertible. (b)(AB) = B A. H.A.O

11. A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order. H.A.O

12. Example 7 Consider the matrices A=B=AB= Applying the formula in Theorem1.4.5, we obtain A = B = (AB) = Also, B A = = Therefore, (AB) = B A as guaranteed by Theorem 1.4.6. 12 1 3 3 2 2 3 7 6 9 8 3 -2 -1 1 1 -1 3232 4 -3 - 9292 7272 1 -1 3232 3 -2 -1 1 4 -3 - 9292 7272 H.A.O

13. POWERS OF A MATRIX Definition. If A is a square matrix, then we define the nonnegative integer powers of A to be Moreover, if A is invertible, then we define the negative integer powers to be Aº = I A ⁿ = AA∙ ∙ ∙A ( n >0) n factors Aˉ ⁿ = ( Aˉ¹ )ⁿ = Aˉ¹ Aˉ¹ ∙ ∙ ∙ Aˉ¹ n factors H.A.O

14. Theorem 1.4.7. If A is a square matrix and r and s are integers, then r s r+s r s rs A A = A (A ) = A H.A.O

15. Theorem 1.4.8. If A is an invertible matrix, then: (a) Aˉ¹ is invertible and ( Aˉ¹ ) ˉ¹ = A. (b) A ⁿ is invertible and ( A ⁿ) ˉ¹ = ( Aˉ¹ ) ⁿ for n=0,1,2,… (c) For any nonzero scalar k, the matrix kA is invertible and ( kA ) ˉ¹ = Aˉ¹. 1k1k H.A.O

16. Example 8. Let A and Aˉ¹ be as in Example 7,that is A = A ˉ¹ = Then A ³ = = A ˉ ³ =( A ˉ¹ ) ³ == 1 2 1 3 3 -2 -1 1 1 2 1 3 1 2 1 3 1 2 1 3 11 30 15 41 3 -2 -1 1 3 -2 -1 1 3 -2 -1 1 41 -30 -15 11 H.A.O

17. POLYNOMIAL EXPRESSIONS INVOLVING MATRICES If A is a square matrix, say m × m, and if p(x)= a + a x + ∙ ∙ ∙+a x (1) is any polynomial, then we define p(A)= a I + a A + ∙ ∙ ∙+a A Where I is the m × m identity matrix. In words, p(A) is the m × m matrix that results when A is substituted for x in (1) and a is replaced by a I. 0 0 1 1 n n n n 00 H.A.O

18. Example 9. If p(x)=2x²-3x+4 and A= Then p ( A )=2A²-3A+4I=2 -3 +4 = - + = -1 2 0 3 -1 2 0 3 2 -1 2 0 3 1 0 0 1 2 8 0 18 -3 6 0 9 4 0 0 4 9 2 0 13 H.A.O

19. Theorem 1.4.9. If the sizes of the matrices are such that the stated operations can be performed, then (a) (( A ) ) = A (b) ( A+B ) = A +B and ( A-B ) = A - B (c) ( kA ) = kA, where k is any scalar (d) ( AB ) = B A TTT T T T TTTTT TT H.A.O

20. The transpose of a product of any number of matrices is equal to the product of their transposes in the reverse order. H.A.O

21. Theorem 1.4.10. If A is an invertible matrix, then A is also invertible and (A )ˉ¹ = (A ˉ¹) (4) TT H.A.O

22. Example 10 Consider the matrices A = Applying Theorem 1.4.5 yields A ˉ¹ = (A )ˉ¹ = As guaranteed by Theorem 1.4.10, these matrices satisfy(4) T T -5 -3 2 1 -5 2 -3 1 1 3 -2 -5 1 -2 3 -5 H.A.O