GATE EC 2014 Set 2 — Question 38
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Engineering Mathematics → Linear Algebra → Determinants
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Question
The maximum value of the determinant among all real symmetric matrices with trace 14 is ________.
Correct answer
48.9 to 49.1
Solution
Let the real symmetric matrix be .
The trace of the matrix is given as .
The determinant of the matrix is .
To maximize the determinant, we must minimize . Since is a real number, the minimum value of is 0 (when ).
Thus, .
Given , the product is maximized when the two numbers are equal (by the AM-GM inequality or simple calculus):
.
The maximum determinant is .
The trace of the matrix is given as .
The determinant of the matrix is .
To maximize the determinant, we must minimize . Since is a real number, the minimum value of is 0 (when ).
Thus, .
Given , the product is maximized when the two numbers are equal (by the AM-GM inequality or simple calculus):
.
The maximum determinant is .
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