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..?

Thanks in advance.. (Bow)

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..?

Thanks in advance.. (Bow)

You must be careful.

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

If you use x=0 you get

$\displaystyle c(0)=\text{det}(-A)=6 \implies (-1)^3\text{det}(A)=6$

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?

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..?

Thanks in advance.. (Bow)

Since the det(xI- A), with x= 0 is just det(0-A). Multiplying one row of a matrix by -1 multiplies the determinant -1. Multiplying A by -1 multiplies all three rows by -1 and so multiplies the determinant by -1.

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

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 $\displaystyle (-1)^4= 1$

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

Thanks **HallsofIvy**.. U are always answering my question perfectly.. (Rofl)