linear-guest

2.2 Review Problems 49
2. Solve the following linear system:
⎛
⎞
1 1 0 1 0 1 0 1
0 0 1 2 0 2 0 −1
⎜0 0 0 0 1 3 0 1
⎟
⎝0 0 0 0 0 2 0 −2⎠ .
0 0 0 0 0 0 1 1
2x 1 + 5x 2 − 8x 3 + 2x 4 + 2x 5 = 0
6x 1 + 2x 2 −10x 3 + 6x 4 + 8x 5 = 6
3x 1 + 6x 2 + 2x 3 + 3x 4 + 5x 5 = 6
3x 1 + 1x 2 − 5x 3 + 3x 4 + 4x 5 = 3
6x 1 + 7x 2 − 3x 3 + 6x 4 + 9x 5 = 9
Be sure to set your work out carefully with equivalence signs ∼ between
each step, labeled by the row operations you performed.
3. Check that the following two matrices are row-equivalent:
( ) ( )
1 4 7 10 0 −1 8 20
and
.
2 9 6 0 4 18 12 0
Now remove the third column from each matrix, and show that the
resulting two matrices (shown below) are row-equivalent:
( ) ( )
1 4 10 0 −1 20
and
.
2 9 0 4 18 0
Now remove the fourth column from each of the original two matrices,
and show that the resulting two matrices, viewed as augmented
matrices (shown below) are row-equivalent:
( 1 4 7
2 9 6
)
and
( 0 −1 8
4 18 12
)
.
Explain why row-equivalence is never affected by removing columns.
4. Check that the system of equations corresponding to the augmented
matrix
⎛
1 4
⎞
10
⎝3 13 9⎠
4 17 20
49