How to find the determinant of matrix if given characteristic equation only?

• Jun 16th 2011, 07:06 AM
nameck
How to find the determinant of matrix if given characteristic equation only?
hey there..

The question is..
lambda = x (i dont know how to type the symbol..)

Let A be n x n square matrix. Suppose that
det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

I know the answer is -6 by substitute lambda@x with 0..
However.. i just want to know.. why should i substitute it with 0..?

• Jun 16th 2011, 07:11 AM
TheEmptySet
Re: How to find the determinant of matrix if given characteristic equation only?
Quote:

Originally Posted by nameck
hey there..

The question is..
lambda = x (i dont know how to type the symbol..)

Let A be n x n square matrix. Suppose that
det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

I know the answer is -6 by substitute lambda@x with 0..
However.. i just want to know.. why should i substitute it with 0..?

You must be careful.

$\displaystyle c(x)=\text{det}(xI-A)=x^3-3x^2+x+6$

If you use x=0 you get

$\displaystyle c(0)=\text{det}(-A)=6 \implies (-1)^3\text{det}(A)=6$
• Jun 16th 2011, 07:16 AM
nameck
Re: How to find the determinant of matrix if given characteristic equation only?
yup2... so.. -(det A) = 6.. so.. det A = -6 right?
ok then.. why it is 0?

Got one more question..
if i've got 4 x 4 matrix,
the sign will be positive?
(-1)^4 det(A) correct?
• Jun 16th 2011, 07:23 AM
HallsofIvy
Re: How to find the determinant of matrix if given characteristic equation only?
Quote:

Originally Posted by nameck
hey there..

The question is..
lambda = x (i dont know how to type the symbol..)

Let A be n x n square matrix. Suppose that
det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

I know the answer is -6 by substitute lambda@x with 0..
However.. i just want to know.. why should i substitute it with 0..?

Yes, if you multiply a 4 by 4 matrix by -1, you have multiplied all 4 rows by -1 and so multiply the determinant by $\displaystyle (-1)^4= 1$