Reducing Matrices with Variables

[JoD]

Hi I'm having trouble reducing these matrices:

 

1 5 a

-4 -5a -16

a 20 16

 

and

 

k 3 4

3 5 8

3 -2 k

 

I can get them both down to 2 zeros in the first column, but can't seem to reduce to one in the second column. Thanks for any help!

[Country Boy]

1 5 a

-4 -5a -16

a 20 16

 

I assume you added four times the first row to the second row and subtracted a times the first row to the third row to get

1 5 a

0 20-5a 4a-16

0 20-5a 16-a^2

 

Do you notice that the numbers in the second and third rows of the second column are the same? I would go ahead and subtract the second row from the third:

1 5 a

0 20-5a 4a-16

0 0 32-4a- a^2

 

Now 20- 5a= -5(a- 4) and 4a-16= 4(a-4) so dividing the second row by 20-5a give 0 1 -4/5. You have now

 

1 5 a

0 1 -4/5

0 0 32-4a- a^2

 

Of course, you can subract 5 times the second row now from the first row:

1 0 a+ 4

0 1 -4/5

0 0 32- 4a- a^2

 

Can you finish?