`Important properties of Eigen values:-`

$(1)$Sum of all eigen values$=$Sum of leading diagonal(principle diagonal) elements=Trace of the matrix.

$(2)$ Product of all Eigen values$=Det(A)=|A|$

$(3)$ Any square diagonal(lower triangular or upper triangular) matrix eigen values are leading diagonal (principle diagonal)elements itself.

Example$:$$A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$

` Diagonal matrix`

Eigenvalues are $1,1,1$

$B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$

`Upper triangular matrix`

Eigenvalues are $1,1,1$

$C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$

`Lower triangular matrix`

Eigenvalues are $1,1,1$

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Now come to the question

$P = \begin{vmatrix} p_{11} \quad p_{12} \\ p_{21} \quad p_{22} \end{vmatrix}_{2\times 2}$ are `non zero`

, and `one of its `

`eigen values is zero.`

Suppose Eigen values are $\lambda_{1}$ and $\lambda_{2}$

Apply the $2^{nd}$ property

$\lambda_{1}.\lambda_{2}=|P|$-----------$>(1)$

`one of its `

`eigen values is zero. So we can take `

$\lambda_{1}=0$

Now from the euation $(1),$we get

$0.\lambda_{2}=|P|$

$\Rightarrow |P|=0$

$\Rightarrow \begin{vmatrix} p_{11} \quad p_{12} \\ p_{21} \quad p_{22} \end{vmatrix}=0$

${\color{DarkGreen}{p_{11}p_{22}-p_{12}p_{21}=0} }$