(a) Consider the following two 2 x 2 matrices: 1 2 A = and B= 2...

    

 

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(a) Consider the following two 2 x 2 matrices: 1 2 A = and B= 2 3 :(. -2-5 9). By evaluating both sides of the equation by hand, verify that: AT (A +2B)-1=2ATBT (b) Now suppose that A and B are arbitrary nxn matrices such that A is invertible. Using general properties of matrix operations (Tables 5.1-5.4 in the Unit Notes), show that AT (A+2B)-1=2ATBT Make sure you show all the steps. 2. [10 marks] Consider the following linear transformations of the plane: T = "rotation through 90 clockwise" T = "horizontal shear with factor 1" T3= "rotation through 90 anti-clockwise" (a) Write down the matrices A1, A2, A3 that correspond to the respective transfor- mations. (b) Use matrix multiplication to determine the combined geometric effect of T followed by T followed by T3. Describe the geometric effect. (c) Use matrix multiplication to determine the combined geometric effect of T followed by T followed by T3, and compare it with the result from (b). 5. [5 marks] Use mathematical induction to prove that if (tn) is a sequence defined recursively by t = 3; tn=tn-1+4n-2, n2, then t=2n+1 for all integers n 1. 6. [3 marks] Use the axioms and fundamental identities of Boolean algebra to simplify the following Boolean expression (showing all the steps): x' (y(x+2))' (a) Consider the following two 2 x 2 matrices: 1 2 A = and B= 2 3 :(. -2-5 9). By evaluating both sides of the equation by hand, verify that: AT (A +2B)-1=2ATBT (b) Now suppose that A and B are arbitrary nxn matrices such that A is invertible. Using general properties of matrix operations (Tables 5.1-5.4 in the Unit Notes), show that AT (A+2B)-1=2ATBT Make sure you show all the steps. 2. [10 marks] Consider the following linear transformations of the plane: T = "rotation through 90 clockwise" T = "horizontal shear with factor 1" T3= "rotation through 90 anti-clockwise" (a) Write down the matrices A1, A2, A3 that correspond to the respective transfor- mations. (b) Use matrix multiplication to determine the combined geometric effect of T followed by T followed by T3. Describe the geometric effect. (c) Use matrix multiplication to determine the combined geometric effect of T followed by T followed by T3, and compare it with the result from (b). 5. [5 marks] Use mathematical induction to prove that if (tn) is a sequence defined recursively by t = 3; tn=tn-1+4n-2, n2, then t=2n+1 for all integers n 1. 6. [3 marks] Use the axioms and fundamental identities of Boolean algebra to simplify the following Boolean expression (showing all the steps): x' (y(x+2))'