Question

Which of the following options is CORRECT for the eigenvalues \(\lambda\) of the given matrix?
\[ \begin{bmatrix} 8 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix} \]

• A. \(\lambda = 1, 1, 1\)
• B. \(\lambda = 8, 5, 7\)
• C. \(\lambda = 0, 0, 0\)
• D. \(\lambda = 40, 1, 35\)

Hint:

For a diagonal matrix, the eigenvalues are simply the entries on the main diagonal.

Answer 1

The Correct Option is B

Step 1: Recall the property of a diagonal matrix:
The given matrix is a diagonal matrix because all the entries outside the main diagonal are zero. For any diagonal matrix, the eigenvalues are simply the entries lying on the main diagonal, since the characteristic equation of a diagonal matrix factors directly from these entries.
Step 2: Write the characteristic equation:
For the matrix \(A = \begin{bmatrix} 8 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}\), the characteristic equation is \(\det(A - \lambda I) = 0\). Since \(A - \lambda I\) is also diagonal, with entries \(8 - \lambda\), \(5 - \lambda\) and \(7 - \lambda\), the determinant is simply the product of these three terms, \((8 - \lambda)(5 - \lambda)(7 - \lambda) = 0\).
Step 3: Solve for the eigenvalues:
Setting each factor equal to zero gives \(\lambda = 8\), \(\lambda = 5\) and \(\lambda = 7\). These are the three eigenvalues of the matrix, and they exactly match the diagonal entries as expected.
Step 4: Check the options:
Option A gives \(\lambda = 1, 1, 1\), which does not match the diagonal entries. Option B gives \(\lambda = 8, 5, 7\), which is exactly what we calculated. Option C gives \(\lambda = 0, 0, 0\), which would only be correct if the matrix were the zero matrix. Option D gives \(\lambda = 40, 1, 35\), which does not correspond to any correct calculation for this matrix.
Final Answer:
\[ \boxed{\lambda = 8, 5, 7} \]