Exercise 7.[Reduced Row Echelon Form]Find the reduced row echelon form of the matrices-22-12033-31-422(b)4-3-4-2(a)-421-4-1-31-4Can you write a computer program to convert am×nmatrix to reduced row echelonform?[100-2010-10010;10020106500185]Exercise 8.[Matrix Operations]LetA=20-110-2,B=-311-311,C=3-1-1-3Whenever possible, perform the following operations. If a computation cannot be made,explain why. [Note:ATmeans the transpose ofA.]Exercise 9.[Matrix Operations](a) IfAis 3-by-5,Bis 2-by-3, and if you stack the matricesA,B,CandDinto abiggerPmatrix asP=ACDB, what should be the dimensions of matrixCandmatrixD?(b) IfAis 3-by-2 andGis 3-by-4, what should the dimensions of matricesBandEbeso that the matrix multiplicationABEGmake sense?(c) IfAis 3-by-2 andBis 3-by-4, what should the dimensions of matricesEandGbeso that the matrix multiplicationABEGmake sense?KVL/Aug19
Exercise 10.[Properties of Determinant](a) IfAis a 3×3 matrix and det(A) = 10, find det(3A), det(2A-1), and det[(2A)-1].(b) IfA=abcdefghiand det(A) = 10, find detagdbhecif(c) List the properties of determinant that you have used in parts (a) and (b).[ 270;4/5;1/80 ]Exercise 11.[Determinant via ERO]Use the method of ERO, find the determinant of the matrix-645282-1-42[-168 ]KVL/Aug19
EE2007/IM2007 Tutorial 3Linear AlgebraElementary Matrices, LU FactorizationExercise 12.[Elementary Matrices]LetEandFbe elementary matrices. IfEadds row 1 to row 2, andFadds row 2 to row1, doesEFequalFE? Justify your answer.Exercise 13.[Elementary Matrices](a) Find the matrixPthat will re-arrange the column vectoru=abcdefTtov=acebdfT.(b) Find the matrixQthat will re-arrange the row vectorp=abctoq=acb.(c) If, in (b), the dimensions ofa,b, andcare 2-by-2, 2-by-3, and 2-by-4 respectively.What will be the dimensions ofQ?Write down the matrixQ.How would youranswer change if the dimensions ofa,b, andcare n-by-2, n-by-3, and n-by-4 respec-tively?
Exercise 14.[Determinants and Inverses of Elementary Matrices]Write down the elementary matrix, its determinant and inverse for each of the followingelementary row operation on a 4×4 matrix?(a) Interchange rows 1 and 3(b) Multiply row 3 by a factor of 5(c) Add eight times of row 2 to row 1[ The determinants are -1, 5 and 1; The inverses are elementary matrices that would”undo” the corresponding elementary row operations ]Exercise 15.[Determinant via Elementary Matrices]Use the method of Elementary Matrices, find the determinant of the matrix-645282-1-42[-168 ]Exercise 16.[Elementary Matrices]LetA=a11a12a13a21a22a23a31a32a33,|A|= 5,andB=001010100nA001010100nwherenis a positive integer.(a) Withn= 1, explain, in terms of row and column operations, what operations wereperformed on matrixA.