Chapter 3

# Chapter 3 - CHAPTER 3 2 Linear Algebra 3.1 1 Matrices Sums...

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CHAPTER 3 Linear Algebra 3.1 Matrices: Sums and Products ! Do They Compute? 1. 2 206 424 202 A = L N M M M O Q P P P 2. A B += L N M M M O Q P P P 2 163 232 103 3. 2 CD , Matrices are not compatible 4. AB = −− L N M M M O Q P P P 133 27 2 131 5. BA = L N M M M O Q P P P 539 212 10 1 6. CD = L N M M M O Q P P P 31 0 81 2 926 7. DC = L N M O Q P 11 67 8. DC () = L N M O Q P T 16 17 9. C D T , Matrices are not compatible 10. D C T , Matrices are not compatible 11. A 2 20 0 211 0 00 2 = L N M M M O Q P P P 12. AD , Matrices are not compatible 13. A I −= L N M M M O Q P P P 3 203 100 14. 43 20 01 0 001 3 BI L N M M M O Q P P P 15. C I 3 , Matrices are not compatible 16. AC = L N M M M O Q P P P 29 03 165

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166 CHAPTER 3 Linear Algebra ! Rows and Columns in Products 17. (a) 5 columns (b) 4 rows (c) 64 × ! Products with Transposes 18. (a) AB T = L N M O Q P =− 14 1 1 3 (b) AB T = L N M O Q P −= L N M O Q P 1 4 11 44 (c) BA T L N M O Q P 1 4 3 (d) BA T = L N M O Q P = −− L N M O Q P 1 1 ! Reckoning 19. The following proofs are carried out for 22 × matrices. The proofs for general nn × matrices follow along the same lines. L N M O Q P L N M O Q P = L N M O Q P +− () = L N M O Q P L N M O Q P = L N M O Q P + L N M O Q P = L N M O Q P = aa bb ab a 11 12 21 22 11 12 21 22 11 11 12 12 21 21 22 22 11 12 21 22 11 12 21 22 11 12 21 22 11 12 21 22 11 11 12 12 21 21 22 22 11 a f a f af ba b 11 12 12 21 21 22 22 L N M O Q P 20. Compare += L N M O Q P + L N M O Q P = ++ L N M O Q P L N M O Q P + L N M O Q P = L N M O Q P baba 11 12 21 22 11 12 21 22 11 11 12 12 21 21 22 22 11 12 21 22 11 12 21 22 11 11 12 12 21 21 22 22 By commutativity of the real numbers, the matrices + and + are the same. 21. cd a cd a a ca da ca da ca da ca da ca ca ca ca da da da da c d + ()=+ L N M O Q P = + + + + L N M O Q P = L N M O Q P = L N M O Q P + L N M O Q P = L N M O Q P + L N M O Q P =+ A AA 11 12 21 22 11 12 21 22 11 11 12 12 21 21 22 22 11 12 21 22 11 12 21 22 11 12 21 22 11 12 21 22 22. cc ca b b b b ca cb ca cb ca cb ca cb ca ca ca ca cb cb cb cb c c + = L N M O Q P = L N M O Q P = L N M O Q P = L N M O Q P + L N M O Q P = L N M O Q P + L N M O Q P 11 11 12 12 21 21 22 22 11 11 12 12 21 21 22 22 11 11 12 12 21 21 22 22 11 12 21 22 11 12 21 22 11 12 21 22 11 12 21 22 a f ! Properties of the Transpose Rather than grinding out the proofs of Problems 23–26, we make the following observations: 23. T T bg = . Interchanging rows and columns of a matrix two times reproduce the original matrix.
SECTION 3.1 Matrices: Sums and Products 167 24. A B A B + () =+ T TT . Add two matrices and then interchange the rows and columns of the resulting matrix. You get the same as first interchanging the rows and columns of the matrices and then adding. 25. kk AA = T T . Demonstrate that it makes no difference whether you multiply each element of matrix A before or after rearranging them to form the transpose. 26. AB B A = T . This identity is not so obvious. Due to lack of space we verify the proof for 22 × matrices. The verification for 33 × and higher-order matrices follows along exactly the same lines.

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## This lab report was uploaded on 04/04/2008 for the course APPM 2360 taught by Professor Williamheuett during the Fall '07 term at Colorado.

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Chapter 3 - CHAPTER 3 2 Linear Algebra 3.1 1 Matrices Sums...

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