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MATH115MF02N - Math 115 2002 Midterm Test Problem 1(a...

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Math 115 2002 Midterm Test Problem 1. (a) Determine the values of a and b for which the following system of linear equations has (i) exactly one solution, (ii) infinitely many solution, and (iii) no solutions. x 1 + ax 3 = 2 x 1 + x 2 + ( a + 1) x 3 = 7 x 1 + x 2 + 2 ax 3 = b + 2 . (b) For the values of a and b in part (ii) above, give the general solution to the system. Problem 2. (a) A 3 × 3 matrix B has the following elementary row operations performed on it, in the order given: 1) R 1 R 1 + aR 2 (add a times row 2 to row 1) 2) R 3 R 2 ( interchange row 2 and row 3) 3) R 2 bR 2 (multiply row 2 by b ) where a and b are non-zero real numbers. Find the matrix A such that the matrix product AB gives the same result as preforming the above three elementary row operations on B . (b) For the matrices A and B from part (a) of this question, if det( B ) = 5, calculate det( AB ). Problem 3. Given that C and D are 5 × 5 matrices with det( C ) = 2 and det( D ) = 3. determine the numerical value of each of the following: (i) det( CD ) (ii) det( C t (iii) det( D - 1 (iv) det(2 D ) (v) det( C 3 ) Problem 4.
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