Answer to Question #294555 in Linear Algebra for Ilyas

Question

Find the product of eigen value of matrix

Answer 1

From the property of eigenvalues, the product of eigenvalues of a matrix equals the determinant of the matrix.

Now, the given matrix is;

$\displaystyle P=\begin{bmatrix} 2 & 0&1 \\ 4 & -3&3\\ 0&2&-1 \end{bmatrix}$, and $\displaystyle |P|=\begin{vmatrix} 2 & 0 & 1 \\ 4 & -3 & 3\\ 0 & 2 & -1 \end{vmatrix}=2\begin{vmatrix} -3 & 3 \\ 2 & -1 \end{vmatrix}+1\begin{vmatrix} 4 & -3 \\ 0 & 2 \end{vmatrix}=2(3-6)+1(8-0)\\\quad\ \ =2(-3)+1(8)=-6+8=2$

Thus, the product of the eigenvalues of $\displaystyle P=\begin{bmatrix} 2 & 0&1 \\ 4 & -3&3\\ 0&2&-1 \end{bmatrix}$is $2.$