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Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E] are symmetric. Following statements are made with respect to these matrices:
(i) Matrix product [F]T [C]T [B] [C] [F] is a scalar.
(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?
(i) Matrix product [F]T [C]T [B] [C] [F] is a scalar.
(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?
- a)Statement (i) is true but (ii) is false.
- b)Statement (i) is false but (ii) is true.
- c)Both the statements are true.
- d)Both the statements are false.
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1,...
Explanation:
To determine the validity of the given statements, let's analyze each statement separately.
(i) Matrix product [F]T[C]T [B] [C][F] is a scalar:
In order for the matrix product [F]T[C]T [B] [C][F] to be a scalar, the dimensions of the resulting matrix should be 1x1, i.e., it should be a scalar value.
Given that [F] is a 5x1 matrix and [C] is a 3x5 matrix, the product [F]T[C]T will result in a 1x3 matrix. The product of this matrix with [B] will be a 1x3 matrix multiplied by a 3x3 matrix, resulting in a 1x3 matrix. Finally, the product of this matrix with [C][F] will be a 1x3 matrix multiplied by a 3x5 matrix, resulting in a 1x5 matrix.
Therefore, the statement (i) is false because the resulting matrix is not a scalar.
(ii) Matrix product [D]T [F] [D] is always symmetric:
In order for the matrix product [D]T [F] [D] to be symmetric, its transpose should be equal to itself.
Given that [D] is a 5x3 matrix and [F] is a 5x1 matrix, the product [D]T [F] will result in a 3x1 matrix. The product of this matrix with [D] will be a 3x1 matrix multiplied by a 5x3 matrix, resulting in a 3x3 matrix.
Taking the transpose of this resulting matrix will give a 3x3 matrix. However, in general, the product of a matrix with its transpose is not necessarily symmetric. It is only symmetric if the original matrix is square and symmetric.
Therefore, the statement (ii) is false because the resulting matrix is not always symmetric.
Hence, both statements (i) and (ii) are false, and the correct answer is option 'D' - Both the statements are false.
To determine the validity of the given statements, let's analyze each statement separately.
(i) Matrix product [F]T[C]T [B] [C][F] is a scalar:
In order for the matrix product [F]T[C]T [B] [C][F] to be a scalar, the dimensions of the resulting matrix should be 1x1, i.e., it should be a scalar value.
Given that [F] is a 5x1 matrix and [C] is a 3x5 matrix, the product [F]T[C]T will result in a 1x3 matrix. The product of this matrix with [B] will be a 1x3 matrix multiplied by a 3x3 matrix, resulting in a 1x3 matrix. Finally, the product of this matrix with [C][F] will be a 1x3 matrix multiplied by a 3x5 matrix, resulting in a 1x5 matrix.
Therefore, the statement (i) is false because the resulting matrix is not a scalar.
(ii) Matrix product [D]T [F] [D] is always symmetric:
In order for the matrix product [D]T [F] [D] to be symmetric, its transpose should be equal to itself.
Given that [D] is a 5x3 matrix and [F] is a 5x1 matrix, the product [D]T [F] will result in a 3x1 matrix. The product of this matrix with [D] will be a 3x1 matrix multiplied by a 5x3 matrix, resulting in a 3x3 matrix.
Taking the transpose of this resulting matrix will give a 3x3 matrix. However, in general, the product of a matrix with its transpose is not necessarily symmetric. It is only symmetric if the original matrix is square and symmetric.
Therefore, the statement (ii) is false because the resulting matrix is not always symmetric.
Hence, both statements (i) and (ii) are false, and the correct answer is option 'D' - Both the statements are false.
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Community Answer
Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1,...
Here, i) write as ,{ [F ]1×5 [c] 5×3 }{[B]3×3[C]3×5[F]5×1}
now , let, [F][c] = [A]
also, [B][c][f]= [Z]
now , we get, {[A]1×3}{[Z]3×1} = [ element ] 1×1
:. Here we have one element in a matrix so is not scalar.
ii) {[D]Ț[F][D]}T = is not symmetric.
because here A' ≠ A
here i) and ii) both is false.
now , let, [F][c] = [A]
also, [B][c][f]= [Z]
now , we get, {[A]1×3}{[Z]3×1} = [ element ] 1×1
:. Here we have one element in a matrix so is not scalar.
ii) {[D]Ț[F][D]}T = is not symmetric.
because here A' ≠ A
here i) and ii) both is false.
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Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer?
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Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. 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 Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer?.
Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. 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 Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer?.
Solutions for Real matrices [A]3x1,[B]3x3, [C]3 x 5 [D]5 x 3, [E]5 x 5 and [F]5 x 1, are given. Matrices [B] and [E]are symmetric. Following statements are made with respect to these matrices:(i) Matrix product [F]T[C]T [B] [C][F] is a scalar.(ii) Matrix product [D]T [F] [D] is always symmetric. With reference to above statements, which of the following applies?a)Statement (i) is true but (ii) is false.b)Statement (i) is false but (ii) is true.c)Both the statements are true.d)Both the statements are false.Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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