Question

Consider a system of linear equations: x−y + 3z + u = 3 2x−2y + 7z + u = 2 x−y + 2z + u = 1 1. Write down the augmented matrix of the system, and take this matrix to the reduced row echelon form. 2. Determine the leading and the free variables of the system, and write down its general solution.

Answer 1

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1. The augmented matrix of the given system of linear equations is A(say) =

1	-1	3	1	3
2	-2	7	1	2
1	-1	2	1	11

The matrix A can be reduced to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row

Add 1 times the 2nd row to the 3rd row

Multiply the 3rd row by -1

Add 1 times the 3rd row to the 2nd row

Add -1 times the 3rd row to the 1st row

Add -3 times the 2nd row to the 1st row

Then the RREF of A is

1	-1	0	0	31
0	0	1	0	-8
0	0	0	1	-4

2. It may be observed that x, z and u are the leading variables and y is a free variable.

The given system of linear equations is equivalent to x-y=31 or, x=31+y, z= -8 and u = -4.Then (x,y,z,u)=(31+y,y,-8,-4) = (31,0,-8,-4)+y(1,1,0,0) = (31,0,-8,-4)+t(1,1,0,0) where t = y is an arbitrary real number. This is the general solution of the given system of linear equations.