linear-guest

326 Sample First Midterm
(e) This amounts to explicitly performing the matrix manipulations
MX − V, MY 1 , and MY 2
to verify that they are all zero vectors.
3. As usual, be sure to write out the row operations above the ∼’s so your work
can be easily checked.
⎛
⎞
1 2 3 4 1 0 0 0
⎜ 2 4 7 11 0 1 0 0
⎟
⎝ 3 7 14 25 0 0 1 0 ⎠
4 11 25 50 0 0 0 1
Check
∼
∼
∼
∼
⎛
⎜
⎝
⎛
⎜
⎝
⎛
⎜
⎝
⎛
⎜
⎝
1 2 3 4 1 0 0 0
0 0 1 3 −2 1 0 0
0 1 5 13 −3 0 1 0
0 3 13 34 −4 0 0 1
⎞
⎟
⎠
1 0 −7 −22 7 0 −2 0
0 1 5 13 −3 0 1 0
0 0 1 3 −2 1 0 0
0 0 −2 −5 5 0 −3 1
1 0 0 −1 −7 7 −2 0
0 1 0 −2 7 −5 1 0
0 0 1 3 −2 1 0 0
0 0 0 1 1 2 −3 1
1 0 0 0 −6 9 −5 1
0 1 0 0 9 −1 −5 2
0 0 1 0 −5 −5 9 −3
0 0 0 1 1 2 −3 1
⎞
⎟
⎠
⎞
⎟
⎠
⎞
⎟
⎠ .
⎛
⎞ ⎛
⎞ ⎛ ⎞
1 2 3 4 −6 9 −5 1 1 0 0 0
⎜2 4 7 11
⎟ ⎜ 9 −1 −5 2
⎟
⎝3 7 14 25⎠
⎝−5 −5 9 −3⎠ = ⎜0 1 0 0
⎟
⎝0 0 1 0⎠ .
4 11 25 50 1 2 −3 1 0 0 0 1
4.
M T M −1 =
( 2 3
1 −1
) ( 1
1
5 5
3
5
− 2 5
) ( 11
= 5
− 4 )
5 .
− 2 5
3
5
326