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All the four entries of $2\times 2$ matrix $P = \begin{vmatrix} p_{11} \quad p_{12} \\ p_{21} \quad p_{22} \end{vmatrix}$ are non zero , and one of its eigen values is zero. Which of the following statements is TRUE?

  1. $p_{11}p_{22} - p_{12}p_{21} = 1$
  2.  $p_{11}p_{22} - p_{12}p_{21} = -1$
  3.  $p_{11}p_{22} - p_{12}p_{21} = 0$
  4.  $p_{11}p_{22} + p_{12}p_{21} = 0$
asked in Others by Boss (15.3k points)
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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 $1,1,1$

--------------------------------------------------------------------------------------------------------------

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

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

$\Rightarrow |P|=0$

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

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

answered by Boss (29.4k points)
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