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A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain a unique solution by multiplying both left and right sides of the equation by AT (the super script T denotes the transpose) and inverting the matrix AT A?
- a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.
- b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.
- c)Yes, can obtain a unique solution provided the matrix AT A is well conditioned
- d)Yes, can obtain a unique solution provided the matrix A is well conditioned.
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A set of linear equations is given in the form Ax = b, where A is a 2 ...
Understanding the Problem
In the equation Ax = b, where A is a 2 × 4 matrix and b is a non-zero vector, the goal is to determine if a unique solution can be found by manipulating the equation.
Matrix Dimensions and Solutions
- A 2 × 4 matrix A has more columns than rows.
- This implies that the system has more variables than equations.
Implications of Matrix A's Dimensions
- Because there are 4 variables (columns) and only 2 equations (rows), the system is underdetermined.
- An underdetermined system typically has infinitely many solutions or no solution.
Multiplying by the Transpose
- When we multiply both sides of the equation by A^T, we get A^T * A * x = A^T * b.
- A^T * A is a 4 × 4 matrix. However, since A has more columns than rows, A^T * A is not guaranteed to be invertible.
Conditions for Unique Solutions
- For A^T * A to be invertible, matrix A must have full row rank (which is not possible here as it cannot have rank 2 with 4 columns).
- Thus, A^T * A will be a singular matrix in most cases.
Conclusion
- Since the system is underdetermined, and A^T * A is not guaranteed to be invertible, it is not possible to obtain a unique solution for any 2 × 4 matrix A.
- Therefore, option B is the correct answer: No, it is not possible to get a unique solution for any 2 × 4 matrix A.
In the equation Ax = b, where A is a 2 × 4 matrix and b is a non-zero vector, the goal is to determine if a unique solution can be found by manipulating the equation.
Matrix Dimensions and Solutions
- A 2 × 4 matrix A has more columns than rows.
- This implies that the system has more variables than equations.
Implications of Matrix A's Dimensions
- Because there are 4 variables (columns) and only 2 equations (rows), the system is underdetermined.
- An underdetermined system typically has infinitely many solutions or no solution.
Multiplying by the Transpose
- When we multiply both sides of the equation by A^T, we get A^T * A * x = A^T * b.
- A^T * A is a 4 × 4 matrix. However, since A has more columns than rows, A^T * A is not guaranteed to be invertible.
Conditions for Unique Solutions
- For A^T * A to be invertible, matrix A must have full row rank (which is not possible here as it cannot have rank 2 with 4 columns).
- Thus, A^T * A will be a singular matrix in most cases.
Conclusion
- Since the system is underdetermined, and A^T * A is not guaranteed to be invertible, it is not possible to obtain a unique solution for any 2 × 4 matrix A.
- Therefore, option B is the correct answer: No, it is not possible to get a unique solution for any 2 × 4 matrix A.
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Community Answer
A set of linear equations is given in the form Ax = b, where A is a 2 ...
AX = B
Where A is 2 × 4 matrices and b ≠ 0
The order of AT is 4 × 2
The order of ATA is 4 × 4
Rank of (A) ≤ min (2, 4) = 2
Rank of (AT) ≤ min (2, 4) = 2
Rank (ATA) ≤ min (2, 2) = 2
As the matrix ATA is of order 4 × 4, to have a unique solution the rank of ATA should be 4.
Therefore, the unique solution of this equation is not possible.
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A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer?
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A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. 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 A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer?.
A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. 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 A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer?.
Solutions for A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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Here you can find the meaning of A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer?, a detailed solution for A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain aunique solutionby multiplying both left and right sides of the equation by AT(the super script T denotes the transpose) and inverting the matrix ATA?a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.c)Yes, can obtain a unique solution provided the matrix ATA is well conditionedd)Yes, can obtain a unique solution provided the matrix A is well conditioned.Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Mathematics tests.
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