When Does the Equation Ax=B Have Unique, Multiple, or No Solutions?

[MathematicalPhysicist]

i need to remember when does Ax=B has a unique solution, more than one, and no solution?

i think that it has a unique solution when A is non singular i.e when det(A) isn't equal zero (at least that's what's written in my notes).

what about more than one solution, what conditipns should be met for either of A,x or b (also for no solution).

i guess that when A isn't non singular then the equation doesn't have solution but I am not sure, forgot about this.
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[doodle]

There is a unique solution if A is full rank, infinite solutions if A is rank-deficient and b lies in the column space (or range) of A, and no solution if A is rank-deficient and b lies in the left-nullspace of A.

The determinant of a matrix is the product of its eigenvalues. A rank-deficient matrix would have some of its eigenvalues equal zero, thus the determinant equals zero. Any matrix A with det(A) = 0 would therefore admit either infinite number of solutions or no solution for Ax = b.

 

[tacman]

You are correct, the equation Ax=B has a unique solution when A is non-singular, meaning its determinant is not equal to zero. This is because a non-singular matrix is invertible, allowing us to solve for x in the equation Ax=B.

For the equation to have more than one solution, there must be infinitely many solutions. This can occur when the rows of A are linearly dependent, meaning one row can be obtained by multiplying another row by a constant. In this case, there are multiple combinations of x that satisfy the equation.

On the other hand, if the rows of A are linearly independent, meaning no row can be obtained by multiplying another row by a constant, then there will be no solution to the equation. This is because there is no way to obtain B by multiplying A with any combination of x values. In this case, we say the equation is inconsistent and has no solution.

It is important to remember the conditions for a unique, multiple, and no solution in linear algebra, as they are fundamental concepts in solving systems of linear equations. I would suggest reviewing these concepts and practicing with some examples to solidify your understanding.