Page 326 - 35Linear Algebra

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
 
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
Check

     
1 2 3 4 −6 9 −5 1 1 0 0 0
2 4 7 11 9 −1 −5 2 0 1 0 0
     
    =   .
 3 7 14 25   −5 −5 9 −3   0 0 1 0 
4 11 25 50 1 2 −3 1 0 0 0 1

4.
!
1 1 11 4
2 3 −
T
−1
M M = 5 5 = 5 2 5 .
3
1 −1 3 − 2 −
5 5 5 5
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