linear algebra consistent

[匿名]

2014-11-01 1:29 am

Determine the conditions satisfied by c and d such that the system of linear equations

[ 1 -2 4 3 3 = [ c d -1 ]^T
3 -2 5 1 4
-6 -4 4 14 2 ] x

is consistent.

Find all solutions of the system when c=1/2 and d=1.

Given answers:
Consistent: -1-6c+4d = 0;
x1= 1/4 -1/2r + s -1/2t
x2= -1/8 + 7/4r + 2s + 5/4t
x3= r
x4= s
x5= t

回答 (1)

用戶10012

2014-11-01 7:33 pm

✔ 最佳答案

( 1 -2 4 3 3 | c )
( 3 -2 5 1 4 | d )
( -6 -4 4 14 2 | -1 )

= ( 1 -2 4 3 3 | c ) R2 - 3R1 to R2 and R3 + 6R1 to R3
( 0 4 -7 -8 -5 | d - 3c )
( 0 -16 28 32 20| -1 + 6c )
= ( 1 -2 4 3 3 | c ) 4R2 + R3 to R3............ (1)
( 0 4 -7 -8 -5 | d - 3c )
( 0 0 0 0 0 | -1 + 6c + 4d - 12c )
For the system of equations to be consistence, - 1 + 6c + 4d - 12c = 0
That is -1 - 6c + 4d = 0
When c = 1/2 and d = 1
Sub into 2nd row of (1), we get
( 0 4 -7 -8 -5 | -1/2 )
Put x5 = t, x4 = s and x3 = r
4x2 - 7r - 8s - 5t = - 1/2
x2 = - 1/8 + 7r/4 + 2s + 5t/4
Sub into 1st row of (1), we get
x1 - 2(-1/8 + 7r/4 +2s + 5t/4) + 4r + 3s + 3t = c = 1/2
so x1 = 1/2 - 1/4 + 7r/2 + 4s + 5t/2 - 4r - 3s - 3t
= 1/4 - r/2 + s - t/2

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