Math 115 2002 Midterm TestProblem 1.(a) Determine the values ofaandbfor which the following system of linear equations has(i) exactly one solution,(ii) infinitely many solution, and(iii) no solutions.x1+ax3=2x1+x2+(a+ 1)x3=7x1+x2+2ax3=b+ 2.(b) For the values ofaandbin part (ii) above, give the general solution to the system.Problem 2.(a) A 3×3 matrixBhas the following elementary row operations performed on it, in the order given:1)R1→R1+aR2(addatimes row 2 to row 1)2)R3R2( interchange row 2 and row 3)3)R2→bR2(multiply row 2 byb)whereaandbare non-zero real numbers.Find the matrixAsuch that the matrix productABgives the same result as preforming theabove three elementary row operations onB.(b) For the matricesAandBfrom part (a) of this question, if det(B) = 5, calculate det(AB).Problem 3.Given thatCandDare 5×5 matrices with det(C) =−2 and det(D) = 3. determine thenumerical value of each of the following:(i) det(CD)(ii) det(Ct(iii) det(D-1(iv) det(2D)(v) det(C3)Problem 4.
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