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?

Question

A set of linear equations is given in the form Ax = b, where A is a 2 &times; 4 matrix with real number entries and b &ne; 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 &times; 4 matrix A.b)No, it is not possible to get a unique solution for any 2 &times; 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?

Answer

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.

Answer

AX = B

Where A is 2 × 4 matrices and b ≠ 0

The order of A^T is 4 × 2

The order of A^TA is 4 × 4

Rank of (A) ≤ min (2, 4) = 2

Rank of (A^T) ≤ min (2, 4) = 2

Rank (A^TA) ≤ min (2, 2) = 2

As the matrix A^TA is of order 4 × 4, to have a unique solution the rank of A^TA should be 4.

Therefore, the unique solution of this equation is not possible.

Similar Mathematics Doubts

Which of the following statements is incorrect regarding the singular solutions of the differential equations?

Ifare three solutions of a non - homogeneous linear differential equationwhere P(x), Q(x) and R(x) are continuous function on [a, b] with a > 0, then itsparticular solution w.r. to the conditions y(0) = 0 y.O) = 1

Consider the differential equation y” + 9y = 0 with the boundary conditions, y(0) = 0, y(2π) = 1, then the differential equation has

Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .

If the rank of an n x n matrix A is (n - 1), then the system of equations Ax= b has

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams