Solving Linear Systems with Cramer's Rule

[lina29]

Homework Statement

Solve the system:
ax1+3x2+x3= y1
ax2= y2
-5x1+ax2-x3=y3

using cramer's rule for those a such that the det of A is not 0

Homework Equations

x1= det(A1)/det(A)
det(A1)=a11+c11+a21c21+a31c31
and A1=

y1 3 1
y2 a 0
y3 a 1

The Attempt at a Solution

By using those equations I got the det(A1) to go ay1+(-3+a)y2-ay3
and the det(A) was a$^{}2$-5a
and x1=(ay1+(-3+a)y2-ay3)/a$^{}2$-5a

But that was counted wrong. I don't understand where I went wrong since I have doublechecked my calculations

 

[I like Serena]

Hi lina29!

You appear to have made a few mistakes with signs.
For starters, det(A) is not a²-5a, but -a²+5a.
How did you calculate it?

And your A1 should be (note the -1 in the lower right corner):
y₁ 3 1
y₂ a 0
y₃ a -1

 

[lina29]

I'm sorry I typed the equation wrong it was
-5x1+ ax2+x3. Even using -a²+5a as the denominator I got it wrong

 

[I like Serena]

Ah, but with your correction your det(A) is still wrong.
Can you recalculate it?

 

[lina29]

and when doing x2=(a-5)y2/(-a²+5a) I also got it counted wrong. Is there something I'm missing out?

 

[lina29]

I'm still getting a²-5a. Using the equation a11c11+a12c12+a13c13.
a11=a
a12=3
a13=1
c11=a
c12=0
c13=-5a

and combining them together I get a²-5a

 

[I like Serena]

Yes, your det(A) is wrong, so x2 is wrong too.
You appear to have made the same mistake calculating det(A2).
Strangely you did calculate det(A1) properly.

 

[lina29]

Could you explain to me where I'm going wrong?

 

[I like Serena]

I'm still getting a²-5a. Using the equation a11c11+a12c12+a13c13.
a11=a
a12=3
a13=1
c11=a
c12=0
c13=-5a

and combining them together I get a²-5a

I don't know this equation for a determinant, but it is not right.

Here's for instance the formula from wikipedia:

 

[I like Serena]

Oh, I think your c11,c12,c13 are the determinants of the sub matrices in the lower 2 rows.
In that case you miscalculated c13.

 

[lina29]

c11, c12, and c13 are determinants of the submatrices. In class we were taught to calculate the determinant using cofactors. Using your method though I had det(A2)= (a-5)y2 -ay3 and det(A)=a²+5a

 

[I like Serena]

c11, c12, and c13 are determinants of the submatrices. In class we were taught to calculate the determinant using cofactors. Using your method though I had det(A2)= (a-5)y2 -ay3 and det(A)=a²+5a

Yep, I got that at second sight.
See my previous post.

So here you have the right det(A), but you still miscalculated det(A2).

 

[lina29]

Got it! (a+5)y2
Thank you so much!

 

[I like Serena]

You're welcome!