Linear Algebra

Section IV. Matrix Operations 227Exerciseš 3.24 Predict the result of each multiplication by an elementary reduction matrix,and then(check) (by multiplying) (it out.) ( ) ( ) ( )3 0 1 2 4 0 1 21 0 1 2(a)(b)(c)0 0 3 4 0 2 3 4 −2 1 3 4( ) ( ) ( ) ( )1 2 1 −1 1 2 0 1(d)(e)3 4 0 1 3 4 1 0̌ 3.25 This table gives the number of hours of each type done by each worker, andthe associated pay rates. Use matrices to compute the wages due.regular overtimewageAlan 40 12regular $25.00Betty 35 6overtime $45.00Catherine 40 18Donald 28 0Remark. This illustrates that in practice we often want to compute linear combinationsof rows and columns in a context where we really aren’t interested in anyassociated linear maps.̌ 3.26 The need to take linear combinations of rows and columns in tables of numbersarises often in practice. For instance, this is a map of part of Vermont and NewYork.SwantonIn part because of Lake Champlain,there are no roads directly connectingsome pairs of towns. For instance,there is no way to go fromWinooski to Grand Isle without goingthrough Colchester. (To simplifythe graph many other roadsand towns have been omitted. Fromtop to bottom of this map is aboutforty miles.)Grand IsleColchesterWinooskiBurlington(a) The incidence matrix of a map is the square matrix whose i, j entry is thenumber of roads from city i to city j. Produce the incidence matrix of this map(take the cities in alphabetical order).(b) A matrix is symmetric if it equals its transpose. Show that an incidencematrix is symmetric. (These are all two-way streets. Vermont doesn’t have manyone-way streets.)(c) What is the significance of the square of the incidence matrix? The cube?3.27 Find the product of this matrix with its transpose.( )cos θ − sin θsin θ cos θ̌ 3.28 Prove that the diagonal matrices form a subspace of M n×n .dimension?What is its3.29 Does the identity matrix represent the identity map if the bases are unequal?