Aliza J.

asked • 08/20/20

Can you please help with eigenvalues of matrix?

From below choices, select all correct approaches for finding the eigenvalues of matrix

LaTeX: A = \left[\begin{array}{rrr}
1&0&4\\
0&2&-1\\
5&1&3
\end{array}\right]

A.) First form a new matrix LaTeX: A - \lambda \cdot I_{3\times 3},  compute its determinant LaTeX: | A - \lambda \cdot I_{3 \times 3}|and then find the roots of the polynomial equation: LaTeX: | A - \lambda \cdot I_{3 \times 3}| = 0.The roots are the eigenvalues.

B.) Use the elementary row or column operations to create two zeros in one row or one column in LaTeX: A

 first, let's call the new matrix LaTeX: B.Compute LaTeX: |B - \lambda \cdot I_{3 \times 3}| and then solve  LaTeX: |B - \lambda \cdot I_{3 \times 3}| = 0 . The roots are the eigenvalues of LaTeX: A.

C.) Form a new matrix LaTeX: A - \lambda \cdot I_{3 \times 3}, then use the elementary row or column operations to create two zeros in one row or one column of this new matrix. Compute LaTeX: |A - \lambda \cdot I_{3 \times 3}| and then solve LaTeX: |A - \lambda \cdot I_{3\times 3}| = 0.The roots are the eigenvalues of LaTeX: A.

D.) The eigenvalues of a square matrix are its diagonal entries so the 3 eigenvalues of LaTeX: A are: LaTeX: 1, \, 2, \, 3.


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