solving matrixs

Question

Use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. (If there is no solution, enter NO SOLUTION. If the system is dependent, express x, y, z, and w in terms of the parameter a.)

x + 2y + z + 3w = 0
x − y + w = 0
y − z + 2w = 0

Answer 1

The augmented matrix is the following

 

⌈1  2  1 3 | 0⌉

 1 -1   0 1 | 0

⌊0  1 -1 2 | 0⌋

 

Subtract the1st row from the 2bd row

⌈1  2  1  3 | 0⌉
 0 -3 -1 -2 | 0
⌊0  1 -1  2 | 0⌋

 

Multiply the 2nd row by -1/3

⌈1 2   1     3 | 0⌉
 0 1 1/3  2/3 | 0
⌊0 1  -1     2 | 0⌋

 

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

Subtract the 2nd row from the 3rd row

⌈1 0   1/3 5/3 | 0⌉
 0 1   1/3 2/3 | 0

⌊0 0 -4/3 4/3 | 0⌋

 

Multiply the 3rd row by -3/4

⌈1 0 1/3 5/3 | 0⌉
0 1  1/3 2/3 | 0
⌊0 0   1   -1 | 0⌋

 

Add -1/3 of the 3rd row to the 1st row

Add -1/3 of the 3rd row to the 2nd row

⌈1 0  0  2  | 0⌉
 0 1  0  1  | 0
⌊0 0 1 -1  | 0⌋

 

This system is now in reduced row echelon form

 

Let w = a

From the 1st row:  x + 2w = 0  => x = -2a

From the 2nd row: y + w = 0    => y = -a

From the 3rd row: z - w = 0     => z = a

 

The solution is (x, y, z, w) = (-2a, -a, a, a)
 

Answer 2

1.  The plural of matrix is matrices.

 

2. You cannot use the graphing utility to write the augmented matrix. But you could put the augmented matrix into the calculator and then use the RREF function on it (TI-84).

 

3. Observe that there are 4 unknowns (x, y, z, w) but only three equations.  But w = y - x and w = ½(z - y). It might be fun to try this by hand, using those facts.