Section 2 Notes - MATH 294 Ex 1.2.11 Solve the system using...

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MATH 294 Ex 1.2.11: Solve the system using Gauss-Jordan elimination. x 1 +2 x 3 +4 x 4 = - 8 x 2 - 3 x 3 - x 4 = 6 3 x 1 +4 x 2 - 6 x 3 +8 x 4 = 0 - x 2 +3 x 3 +4 x 4 = - 12 the augmented matrix -----------------→ 1 0 2 4 . . . - 8 0 1 - 3 - 1 . . . 6 3 4 - 6 8 . . . 0 0 - 1 3 4 . . . - 12 -→ - 3 × (1 st eq ) 1 0 2 4 . . . - 8 0 1 - 3 - 1 . . . 6 0 4 - 12 - 4 . . . 24 0 - 1 3 4 . . . - 12 -→ - 4 × (2 nd eq ) +2 nd eq 1 0 2 4 . . . - 8 0 1 - 3 - 1 . . . 6 0 0 0 0 . . . 0 0 0 0 3 . . . - 6 - 4 × (m4 th eq ) +m4 th eq -→ ÷ 3 1 0 2 0 . . . 0 0 1 - 3 0 . . . 4 0 0 0 0 . . . 0 0 0 0 1 . . . - 2 We get reduced row-echelon form (rref) of the initial matrix. Now we can easily write down the solution. Let’s put x 3 = t where t R (any real number). Then the solution is ( x 1 , x 2 , x 3 , x 4 ) = ( - 2 t, 4 + 3 t, t, - 2), t R . Ex 1.2.25 Suppose matrix A is transformed into matrix B by a sequence of elementary row operations. Is there a sequence of elementary row operations that transforms B into A? Solution: The answer is Yes, there is. We always have a log file containing the sequence of all elementary row operations. In order to get the original matrix A one first needs to invert the order of all operations in the log file and then execute the inverse operations. If an operation is the swapping of i th and j th rows then its inverse is the swapping of j th and i th rows. If new row y new = a · x + y old then old row y old = y new -
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