Below, five systems of linear equations have been put in reduced row echelon for

Question

Below, five systems of linear equations have been put in reduced row echelon form. Identify how many solutions each one has. and enter that number in the blank. Enter the word "infinite" (without the quote marks) if there are infinitely many solutions. [1 0 0 0 01 0 0 0 0 1 0|-2 -2 -1 0] [1 0 0 0 0 0 0 1 0 0 0 0 4 -3 0 0 0 0 2 2 0 0 0 0 4 -5 0 0 0 0 -2 -3 0 0 0 0 |4 5 0 0 0 0] [1 0 0 0 0 0 0 1 0 2 1 0|5 3 1] [1 0 0 0 0 -2 0 0 0 0 -2 0 0 0 0 -3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 |-5 -5 3 0 0] [1 0 0 0 0 0 0 0 1 0 0 0 0 4 3 0 0 0 0 0 0 1 0 0 0 -2 1 5 0 0 0 0 0 0 1 0 0 -5 1 1 -3 0 0]

Explanation / Answer

(a) If X = (x1, x2, x3, x4 )T, then the given matrix equation is equivalent to the linear system x1 = -2, x2 = -2, x3 = -1. x4 is indeterminate. Thus, the solution is X= ( -2, -2,-1, t) where x4 = t , an arbitrary real number. Thus, there are INFINITE solutions.

(b) If X = (x1, x2, x3, x4,x5 , x6 )T, then the given matrix equation is equivalent to the linear system x1 +4x3+ 2x4+4x5 - 2x6 = 4, x2 -3x3 +2x4 -5x5 -3x6 = 5. This is a system of 2 linear equations in 6 variables.Thus, there are INFINITE solutions.

(c) The givem matrix equation is INCONSISTENT and there are no solutions as 0 cannot be equal to 1.

(d) If X = (x1, x2, x3, x4,x5 , x6 )T, then the given matrix equation is equivalent to the linear system x1-2x2 -2 x3 - 3x4 =- 5, x5 = - 5 and x6 =3 This is a system of 3 linear equations in 6 variables.Thus, there are INFINITE solutions.

(e) If X =(x1, x2, x3, x4,x5 , x6 )T, then the given matrix equation is equivalent to the linear system x1+4x3 -2x5 = -5, x2 +3x3+ x5 = 1, x4 + 5x5 =1 and x6 = -3 . This is a system of 4 linear equations in 6 variables.Thus, there are INFINITE solutions.

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