Question

the product of the eigen values of matrix {5 4
1 2} is

Answer 1

Answer:

digit 5412 is a digit

Step-by-step explanation:

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Answer 2

Answer:

The product of eigen values of the given matrix is 6.

Step-by-step explanation:

Eigenvalue:

• For an $n\times n$ matrix A, an eigenvalue is a number λ such that $A\psi = \lambda\psi$ for some nonzero vector $\psi$.
• An $n\times n$ matrix can have n eigenvalues.
• The sum of the n eigenvalues of the matrix A is equal to the trace of matrix.
• The product of the n eigenvalues of A is equal to the determinant of the matrix A.

Given the matrix,

$A=\left[\begin{array}{cc}5&4\\1&2\end{array}\right]$

The eigen values of A are given by

$|A-\lambda I|=0$

where I is an identity matrix.

$\bigg|\left(\begin{array}{cc}5&4\\1&2\end{array}\right)-\lambda\left(\begin{array}{cc}1&0\\0&1\end{array}\right)\bigg|=0$

$\implies \left|\begin{array}{cc}5-\lambda&4\\1&2-\lambda\end{array}\right|=0$

$\implies (5-\lambda)(2-\lambda)-4=0$

$\implies \lambda^{2}-7\lambda+ 6=0$

$\implies (\lambda-6)(\lambda-1)=0$

$\implies \lambda=6~\mbox{or}~\lambda=1$

The eigenvalues are 6 and 1 and their product is 6.

Or else in a simple way:

The determinant of the given matrix is

$det\left[\begin{array}{cc}5&4\\1&2\end{array}\right]=\left|\begin{array}{cc}5&4\\1&2\end{array}\right|$

                    $=5\times 2-4\times 1=10-4=6$

Therefore, the product of eigen values of the given matrix is 6.

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