nrev - Review Problems for Exam One 1 Represent the...

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Review Problems for Exam One 1. Represent the following system of equations by an augmented matrix, reduce this matrix to upper triangular form by Gaussian Elimination and then use back- substitution to find the solution. x + 2 y + 4 z = 0 3 x + 8 y + 8 z = 2 2 x + 6 y + 9 z = 7 2. Put the matrix A = 1 2 0 0 3 2 4 1 3 0 2 4 2 3 1 into reduced row echelon form and then give the general solution to the equation A -→ x = -→ 0 and express in parametric form. 3. Let A be an m × n matrix, where m < n . Explain why the homogeneous system of equation A -→ x = -→ 0 has a non-trivial solution. 4. Let A be an m × n matrix with entries a ij , let B be an n × r matrix with entries b ij , and let C be an n × r matrix with entries c ij . Prove that A ( B + C ) = AB + AC . 5. Prove that if A is invertible, then A T is also invertible. 6. Suppose that -→ u is a solution of A -→ x = -→ 0 and that -→ v is a solution of A -→ x = -→ c . Explain carefully why 2 -→ u + -→ v is a solution of A -→ x = -→ c .

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