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quiz1_S06 solutions

# quiz1_S06 solutions - Princeton University Department of...

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Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 1 – Spring 2006 SOLUTIONS (1) Solve the following system: x 1 + 2 x 2 + x 3 + x 4 = 1 2 x 1 + 2 x 2 - x 3 - x 4 = 0 x 1 - 2 x 2 - 5 x 3 + x 4 = 0 What is the rank of the corresponding coefficient matrix and what is the rank of the corresponding augmented matrix? Solution: We apply Gauss-Jordan elimination to the system written in matrix notation: 2 4 1 2 1 1 2 2 - 1 - 1 1 - 2 - 5 1 ˛ ˛ ˛ ˛ ˛ ˛ 1 0 0 3 5 2 4 1 2 1 1 0 - 2 - 3 - 3 0 - 4 - 6 0 ˛ ˛ ˛ ˛ ˛ ˛ 1 - 2 - 1 3 5 2 4 1 0 - 2 - 2 0 1 3 2 3 2 0 0 0 6 ˛ ˛ ˛ ˛ ˛ ˛ - 1 1 3 3 5 2 4 1 0 - 2 0 0 1 3 2 0 0 0 0 1 ˛ ˛ ˛ ˛ ˛ ˛ 0 1 4 1 2 3 5 The last matrix shows that the solutions of the system are the ( x 1 , x 2 , x 3 , x 4 ) satisfying 8 < : x 1 - 2 x 3 = 0 x 2 + 3 2 x 3 = 1 4 x 4 = 1 2 Taking x 3 as a free variable, the solutions form the vectors 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 2 - 3 2 1 0 3 7 7 5 t + 2 6 6 4 0 1 4 0 1 2 3 7 7 5 , t R . The rank of a matrix is the number of leading ones in its reduced row- echelon form (rref). The leading ones are boldfaced above. As there are three leading ones in the rref of the coefficient matrix and in the rref of the augmented matrix, we see that both of these matrices have rank 3.

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quiz1_S06 solutions - Princeton University Department of...

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