Question

Find the sum and product of the Eigenvalues of the matrix
1 2 3
2 2 4 .
1 2 7
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Answer 1

$\large\underline{\sf{Solution-}}$

The given matrix is

$\rm :\longmapsto\:A \: = \: \begin{gathered}\sf \left[\begin{array}{ccc}1&2&3\\2&2&4\\1&2&7\end{array}\right]\end{gathered}$

We know,

The sum of eigen values is sum of trace of the matrix

We further know,

Trace of a square matrix A, is represented as tr(A) and is defined as sum of diagonal elements.

So, for the given matrix,

$\rm :\longmapsto\:tr(A) = 1 + 2 + 7$

$\bf\implies \:tr(A) = 10$

Hence,

$\red{\boxed{ \bf{ \: Sum \: of \: eigen \: values = tr(A) = 10}}}$

Now,

Further we know that,

The product of eigen values of matrix A is equals to | A |.

So,

Let first find | A |.

$\rm \: |A| = \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}1&2&3\\2&2&4\\1&2&7\end{array}\right | \end{gathered}$

$\rm \: = \: \:1\begin{array}{|cc|}\sf 2 &\sf 4 \\ \sf 2 &\sf 7 \\\end{array} - 2\begin{array}{|cc|}\sf 2 &\sf 4 \\ \sf 1 &\sf 7 \\\end{array} + 3\begin{array}{|cc|}\sf 2 &\sf 2 \\ \sf 1 &\sf 2 \\\end{array}$

$\rm \: = \: \:(14 - 8) - 2(14 - 4) + 3(4 - 2)$

$\rm \: = \: \:6 - 2(10) + 3(2)$

$\rm \: = \: \:6 - 20 + 6$

$\rm \: = \: \: - 8$

$\bf\implies \: |A| = - 8$

Hence,

$\red{\boxed{ \bf{ \: Product \: of \: eigen \: values = |A| = - \: 8}}}$

Additional Information :-

Question :- Prove that the square matrix A and its transpose have same eigen values.

Solution :-

Since A is a square matrix.

We know, its characteristic polynomial is given | A - k I |

Now,

Characteristic polynomial of A' is given by

$\rm \: = \: \: |A' - kI|$

$\rm \: = \: \: |A' - kI'|$

$\rm \: = \: \: |A' - (kI)'|$

$\rm \: = \: \: |(A - kI)'|$

$\rm \: = \: \: |A - kI|$

Thus,

Characteristic polynomial of A and A' is same.

It implies, they have same eigen values.

Answer 2

Answer:

Step-by-step explanation:

he given matrix is

We know,

The sum of eigen values is sum of trace of the matrix

We further know,

Trace of a square matrix A, is represented as tr(A) and is defined as sum of diagonal elements.

So, for the given matrix,

Hence,

Now,

Further we know that,

The product of eigen values of matrix A is equals to | A |.

So,

Let first find | A |.

Hence,