Determining Values for a Given Matrix - How Can I Simplify This Process?

[dracolnyte]

Homework Statement

detA = 0
Matrix A =

| (x+5) 4 4 |
| -4 (x-3) -4 |
| -4 -4 (x-3)|

The Attempt at a Solution

I know that if one row or column is equal to another, then detA = 0, so using the last 2 rows, i can find out that x has to be -1 for row2 and row3 to be equal for detA = 0.

but the answer at the back says x = -1 or 3, how can i solve the 3? I have tried to reduce it to the triangular form, but it got way too messy to be correct. I also tried using the 3x3 matrix trick where you copy the first 2 rows and make a 4th and 5th row out of them and solve for the determinant, also got pretty messy.

Is there some rule that i missed out that can make my life easier on solving this question?

 

[Office_Shredder]

Do you know the expansion by rows trick?

Definitely the best way to do this is to calculate the determinant and then solve the polynomial... trying to find linear dependence conditions is too error prone and you can miss solutions

 

[dracolnyte]

you mean the one the one where you the multiply the diagonals and the subtract it by the other diagonals? in other words
(x+5) 4 4
-4 (x-3) -4
-4 -4 (x-3)
------------
(x+5) 4 4
-4 (x-3) -4

 

[dracolnyte]

a11a22a33 - a11a23a32 - a12a21a33 + a12a23a31 + a13a21a32 - a13a22a31 = 0
it gets really messy with like = x^3 - 11x^2 + 55x - 93

 

[Mark44]

a11a22a33 - a11a23a32 - a12a21a33 + a12a23a31 + a13a21a32 - a13a22a31 = 0

Either your formula isn't right, or you have made in error in calculation.
I worked it out and got a different polynomial, which when factored and set to zero, had roots equal to 3 and -1.

it gets really messy with like = x^3 - 11x^2 + 55x - 93

 

[dracolnyte]

ya sorry, i realized and i did it again, i got x = 3 and -1. my bad, it must be getting late