(3) Consider the following system of equations: egincases x...

(3) Consider the following system of equations: egin{cases} x + y + z = 2 \ x + 3y + 3z = 0 \ x + 3y + 6z = 3 end{cases} (*) (a) Use Gaussian elimination to put the augmented coefficient matrix into row echelon form. The result will be egin{bmatrix} 1 & 1 & 1 & a \ 0 & 1 & 1 & b \ 0 & 0 & 1 & c end{bmatrix} where a = __, b = __, and c = __. (b) Use Gauss-Jordan reduction to put the augmented coefficient matrix in reduced row echelon form. The result will be egin{bmatrix} 1 & 0 & 0 & d \ 0 & 1 & 0 & e \ 0 & 0 & 1 & f end{bmatrix} where d = __, e = __, and f = __. (c) The solutions of (*) are x = __, y = __, and z = __.

Transcript

00:01 Okay, so we have this system of equations, and the first thing we want to do is use gaseum elimination to put the augmented coefficient matrix in row echelon form.

00:10 So if we first construct the augmented matrix, the first row is just going to be the coefficients we have in the first equation.

00:19 So 1, 1, 1, and then 2.

00:22 The second row is going to be 1, 3, 3, 0, and finally we have 1, 3, 6, 3.

00:31 Okay, so the first operation i'm going to do is what i'm going to leave the first equation alone, the first row alone.

00:40 I'm also going to leave the second row alone like this.

00:45 And then from the third row, i'm going to subtract the second row.

00:49 So the third row is going to become the third row minus the second row.

00:53 Well, here we have 1 minus 1, so we get 0.

00:56 Here we have 3 minus 3, so we also get 0.

00:59 Here we have 6 minus 3, so we get 3, and then we get 3 minus 0, which is 3.

01:08 And then the next step is to leave the first row alone, and now i'm going to do two operations in 1, because they're fairly straightforward.

01:17 I'm going to change the second row to the second row minus the first row, so i'm going to do this row minus this row, so we get 1 minus 1 here, so this is just 0, then we get 3 minus 1.

01:31 Minus 1, which is 2.

01:33 And then we get 3 minus 1, which is 2.

01:36 And then we get 0 minus 2, which is minus 2.

01:41 And then the final row, i'm going to just divide everything by 3.

01:45 So we get 0 -0 -1 -1.

01:49 And then the next step is to leave the first row alone still...

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