A. MillerM340 Exam 1February 24, 1993There is a total of 100 points. Put your answers on separatesheets of paper. Keep this exam, hand in the answers and all worksheets.1. (6 points) LetA ="1 21 1#B ="1 −3 −1 21 2 3 −1#C ="1 1−1 0#(a) What is A + C?(b) What is 3A?(c) What is AB?(d) What is BT?(e) What is col3(B)?(f) What is entry2,3(B)?2. (4 points) (a) What is I4×4? (b) What is Z3×5?3. (8 points) Find a matrix R in Echelon form and an invertible matrix Msuch that MA = R whereA =1 20 −13 61 2.4. (6 points) Consider the following three equations: x1+ x2+ x3+ x4= 1,x2+ x3+ x4= 2, and x4+ x5= 3.1(a) Find matrices A and B such that these three equations are equivalentto the single matrix equation AX = B.(b) Find two solutions of this system such that neither is scaler multipleof the other.5. (10 points) Find the inverse of the following matrix:1 0 0 01 1 1 0−1 0 1 10 0 1 06. (4 points) Find a 5 × 5 elementary matrix E such that for any 5 × nmatrix A, EA is the matrix obtained from A by swaping third row and fifthrow of A.7. (10 points) LetA =1 2 31 1 1−1 0 1.Find a 3 × 3 matrix B which is not the zero matrix Z but such that AB = Z.8. (10 points) LetA =0 1 10 1 01 1 0.Write A−1as the product of elementary matrices.9. (4 points) Suppose A and B are 5×5 matrices, det(A) = 3 and det(B) = 7.2What is det(A−1BT)? If it is impossible to be determined by this information,say so.10. (4 points) Suppose C is a 5 × 5 matrix with det(C) = 3. What isdet(2C)? If it is impossible to be determined by this information, say so.11. (4 points) Suppose A and B are 5× 5 matrices, det(A) = 3 and det(B) =−3. What is det(A + B)? If it is impossible to be determined by thisinformation, say so.12. (6 points) What is the det(A) whereA =0 1 11 0 11 1 013. (12 points) Let A be an m × n matrix, B be an n × p matrix.(a) State the definition of C = AB.(b) Let c be a scalar. Prove that A(cB) = (cA)B.14. (12 points) Let A and B be n × n matrices such that(A + B)2= A2+ 2AB + B2.Prove that AB = BA.Hint: What is the simpliest formula you can get for (A + B)2without theassumption that AB =