From the property of eigenvalues, the product of eigenvalues of a matrix equals the determinant of the matrix.
Now, the given matrix is;
P = [ 2 0 1 4 − 3 3 0 2 − 1 ] \displaystyle
P=\begin{bmatrix}
2 & 0&1 \\
4 & -3&3\\
0&2&-1
\end{bmatrix} P = ⎣ ⎡ 2 4 0 0 − 3 2 1 3 − 1 ⎦ ⎤ , and ∣ P ∣ = ∣ 2 0 1 4 − 3 3 0 2 − 1 ∣ = 2 ∣ − 3 3 2 − 1 ∣ + 1 ∣ 4 − 3 0 2 ∣ = 2 ( 3 − 6 ) + 1 ( 8 − 0 ) = 2 ( − 3 ) + 1 ( 8 ) = − 6 + 8 = 2 \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 ∣ P ∣ = ∣ ∣ 2 4 0 0 − 3 2 1 3 − 1 ∣ ∣ = 2 ∣ ∣ − 3 2 3 − 1 ∣ ∣ + 1 ∣ ∣ 4 0 − 3 2 ∣ ∣ = 2 ( 3 − 6 ) + 1 ( 8 − 0 ) = 2 ( − 3 ) + 1 ( 8 ) = − 6 + 8 = 2
Thus, the product of the eigenvalues of P = [ 2 0 1 4 − 3 3 0 2 − 1 ] \displaystyle
P=\begin{bmatrix}
2 & 0&1 \\
4 & -3&3\\
0&2&-1
\end{bmatrix} P = ⎣ ⎡ 2 4 0 0 − 3 2 1 3 − 1 ⎦ ⎤ is 2. 2. 2.
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