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The sum of th of the eigen values of the matrix given below is
[[1, 1, 3], [1, 5, 1], [3, 1, 1]]
(A) 5
(C) 9
(B) 7
(D) 18?
[[1, 1, 3], [1, 5, 1], [3, 1, 1]]
(A) 5
(C) 9
(B) 7
(D) 18?
Most Upvoted Answer
The sum of th of the eigen values of the matrix given below is[[1, 1, ...
Understanding Eigenvalues
The eigenvalues of a matrix provide essential insights into its properties, such as stability and behavior under transformations. The sum of the eigenvalues can be derived from the trace of the matrix.
Calculating the Trace
The trace of a matrix is defined as the sum of its diagonal elements. For the given matrix:
\[
A = \begin{bmatrix}
1 & 1 & 3 \\
1 & 5 & 1 \\
3 & 1 & 1
\end{bmatrix}
\]
- The diagonal elements are:
- \(1\) (from the first row)
- \(5\) (from the second row)
- \(1\) (from the third row)
Sum of the Diagonal Elements
Calculating the trace:
- Trace \( (A) = 1 + 5 + 1 = 7 \)
Eigenvalue Properties
- The sum of the eigenvalues of a matrix equals the trace.
- Hence, the sum of the eigenvalues of matrix \(A\) is \(7\).
Conclusion
The answer to the problem is:
- (B) 7
This value is derived directly from the properties of the matrix and its trace, confirming the relationship between the trace and the sum of the eigenvalues.
The eigenvalues of a matrix provide essential insights into its properties, such as stability and behavior under transformations. The sum of the eigenvalues can be derived from the trace of the matrix.
Calculating the Trace
The trace of a matrix is defined as the sum of its diagonal elements. For the given matrix:
\[
A = \begin{bmatrix}
1 & 1 & 3 \\
1 & 5 & 1 \\
3 & 1 & 1
\end{bmatrix}
\]
- The diagonal elements are:
- \(1\) (from the first row)
- \(5\) (from the second row)
- \(1\) (from the third row)
Sum of the Diagonal Elements
Calculating the trace:
- Trace \( (A) = 1 + 5 + 1 = 7 \)
Eigenvalue Properties
- The sum of the eigenvalues of a matrix equals the trace.
- Hence, the sum of the eigenvalues of matrix \(A\) is \(7\).
Conclusion
The answer to the problem is:
- (B) 7
This value is derived directly from the properties of the matrix and its trace, confirming the relationship between the trace and the sum of the eigenvalues.
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The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18?
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The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18?.
The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of th of the eigen values of the matrix given below is[[1, 1, 3], [1, 5, 1], [3, 1, 1]](A) 5(C) 9(B) 7(D) 18?.
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