Realtion between eigenvalues and determinant of symmetric matrix

Question

$A$ is a $n\times n$ real symmetric matrix with eigen values $a_1,\cdots,a_n$ then
which of the following statements true??

1. product of all eigen values < det(A)
2. product of all eigen values > det(A)
3. product of all eigen values = det(A)
4. If det($A$) = 1 then all eigen values of $A$ are equal to 1.

I tried trial and error method of $2\times 2$ symmetric matrix I get both option 2 and option 3 ??
But I am not sure what is the answer??

Answer 1

Eigenvalues are the roots of $$ P(\lambda)= \det (\lambda I - A) = (\lambda - \lambda_1)(\lambda - \lambda_2)...(\lambda-\lambda_n)$$

Let $\lambda = 0$ and you get $$(-1)^n \det(A) = (-1)^n \lambda_1 \lambda _2...\lambda_n$$ That is $$\det(A) = \lambda_1 \lambda _2...\lambda_n$$ Thus option $3$ is the correct one.