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Let A be the matrix of quadratic form (x1- x2 + 2x3)2.
Then, trace of A is
Then, trace of A is
- a)2
- b)4
- c)6
- d)0
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A...
We are given that A be a matrix of quardratic form (x1-x2 + 2x3)2
or x12 + x22 + 4x32 - 2x1x2 - 4x2x3 + 4x3x1
trace (A) = sum of coefficient of
x12, x22 and x32.
= 1 + 1 + 4
= 6
or x12 + x22 + 4x32 - 2x1x2 - 4x2x3 + 4x3x1
trace (A) = sum of coefficient of
x12, x22 and x32.
= 1 + 1 + 4
= 6
Most Upvoted Answer
Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A...
Quadratic Form:
A quadratic form is a homogeneous polynomial of degree 2 in the variables x1, x2, x3, etc. It can be represented as a matrix equation:
Q(x) = x^T * A * x
where x is a vector (x1, x2, x3, etc.) and A is a symmetric matrix.
Matrix of Quadratic Form:
To find the matrix A of the given quadratic form (x1 - x2 + 2x3)^2, we expand the expression and collect the coefficients of the variables:
Q(x) = (x1 - x2 + 2x3)^2
= x1^2 - 2x1x2 + 4x1x3 - 2x1x2 + x2^2 - 4x2x3 + 4x1x3 - 4x2x3 + 4x3^2
= x1^2 + x2^2 + 4x3^2 - 4x1x2 - 8x2x3 + 8x1x3
From this expression, we can see that the matrix A is:
A = [[1, -2, 4],
[-2, 1, -4],
[4, -4, 4]]
Trace of a Matrix:
The trace of a square matrix is the sum of its diagonal elements. In this case, the matrix A is a 3x3 symmetric matrix.
So, the trace of A is given by:
trace(A) = a11 + a22 + a33
where aij represents the elements of A.
Calculating the Trace:
Let's calculate the trace of A:
trace(A) = 1 + 1 + 4
= 6
Therefore, the correct answer is option 'C', which is 6.
A quadratic form is a homogeneous polynomial of degree 2 in the variables x1, x2, x3, etc. It can be represented as a matrix equation:
Q(x) = x^T * A * x
where x is a vector (x1, x2, x3, etc.) and A is a symmetric matrix.
Matrix of Quadratic Form:
To find the matrix A of the given quadratic form (x1 - x2 + 2x3)^2, we expand the expression and collect the coefficients of the variables:
Q(x) = (x1 - x2 + 2x3)^2
= x1^2 - 2x1x2 + 4x1x3 - 2x1x2 + x2^2 - 4x2x3 + 4x1x3 - 4x2x3 + 4x3^2
= x1^2 + x2^2 + 4x3^2 - 4x1x2 - 8x2x3 + 8x1x3
From this expression, we can see that the matrix A is:
A = [[1, -2, 4],
[-2, 1, -4],
[4, -4, 4]]
Trace of a Matrix:
The trace of a square matrix is the sum of its diagonal elements. In this case, the matrix A is a 3x3 symmetric matrix.
So, the trace of A is given by:
trace(A) = a11 + a22 + a33
where aij represents the elements of A.
Calculating the Trace:
Let's calculate the trace of A:
trace(A) = 1 + 1 + 4
= 6
Therefore, the correct answer is option 'C', which is 6.
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Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer?
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Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer?.
Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be the matrix of quadratic form (x1- x2 + 2x3)2.Then, trace of A isa)2b)4c)6d)0Correct answer is option 'C'. Can you explain this answer?.
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