Computer Science Engineering (CSE) Exam  >  Computer Science Engineering (CSE) Questions  >  Which one of the following statements is TRUE... Start Learning for Free
Which one of the following statements is TRUE about every  matrix with only real eigenvalues?
  • a)
    If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
  • b)
    If the trace of the matrix is positive, all its eigenvalues are positive.
  • c)
    If the determinant of the matrix is positive, all its eigenvalues are positive.
  • d)
    If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Which one of the following statements is TRUE about every matrix with ...
Trace is the sum of all diagonal elements of a square matrix.
Determinant of a matrix = Product of eigen values.
A) Is the right answer. To have the determinant negative ,atleast one eigen value has to be negative(but reverse may not be true). {you can take simple example with upper or lower triangular matrices. In the case option (b) , (c) and (d) reverse is always true .}
View all questions of this test
Most Upvoted Answer
Which one of the following statements is TRUE about every matrix with ...
Understanding Eigenvalues and Matrix Properties
Eigenvalues of a matrix provide insight into its properties, including stability and behavior under transformations. For matrices with real eigenvalues, certain relationships between the trace, determinant, and eigenvalues can be established.

Analyzing Option A: Trace and Determinant
- The **trace** of a matrix is the sum of its eigenvalues.
- The **determinant** is the product of its eigenvalues.
Given:
- **Trace (T)** > 0
- **Determinant (D)** < />
From these conditions:
- The positive trace implies that the sum of the eigenvalues is positive.
- The negative determinant indicates that the product of the eigenvalues is negative.

Implications of these Conditions
- If the product of the eigenvalues is negative (D < 0),="" it="" means="" at="" least="" one="" eigenvalue="" must="" be="" negative.="" this="" is="" because="" the="" product="" of="" an="" odd="" number="" of="" positive="" numbers="" cannot="" yield="" a="" negative="" />
- The positive trace suggests that the positive eigenvalues can compensate for the negative eigenvalue(s) to yield a positive sum.
Therefore, it is guaranteed that at least one eigenvalue is negative, making option A true.

Contrasting with Other Options
- **Option B**: Positive trace does not guarantee all eigenvalues are positive; one can be negative.
- **Option C**: A positive determinant does not guarantee all eigenvalues are positive; for example, one can be negative.
- **Option D**: A positive product of trace and determinant does not ensure all eigenvalues are positive, as seen in specific cases.
Thus, option A stands out as the only universally valid statement regarding real eigenvalues in a matrix.
Explore Courses for Computer Science Engineering (CSE) exam

Similar Computer Science Engineering (CSE) Doubts

Top Courses for Computer Science Engineering (CSE)

Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer?
Question Description
Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer?.
Solutions for Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE). Download more important topics, notes, lectures and mock test series for Computer Science Engineering (CSE) Exam by signing up for free.
Here you can find the meaning of Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice Which one of the following statements is TRUE about every matrix with only real eigenvalues?a)If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.b)If the trace of the matrix is positive, all its eigenvalues are positive.c)If the determinant of the matrix is positive, all its eigenvalues are positive.d)If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice Computer Science Engineering (CSE) tests.
Explore Courses for Computer Science Engineering (CSE) exam

Top Courses for Computer Science Engineering (CSE)

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev