Square matrices, determinants and consistency

[Asif]

This is the first time I'm posting (or rather asking) anything here. I'm a student of elementary linear algebra, therefore please excuse me if my questions come across as dumb or if I make any mistakes:

I have a question about determinants and whether or not a solution exists, etc. I will be focusing on square matrices only:

If the determinant of a matrix is not equal to zero, then does that mean the matrix has a unique solution?

If the determinant is equal to zero, then either the matrix has infinitely many solutions or no solution, correct?

And if it is a homogeneous system, then the system has infinitely many solutions if the determinant is equal to zero, correct?

Thanking you in advance,
Bye.

P.S. Could someone kindly tell me what is meant by singular and non-singular matrices?

 

[matt grime]

One doesn't say a matrix has a solution. Matrices are used to describe sets of linear equations but that is not all they are there for. Second, non-square matrices do not have a determinant.

Non-zero determinant means the matrix is invertible (non-singular), so if you're solving Ax=b, then the answer is given by x=A^{-1}b and the solution is unique.

If the matrix has determinant zero it is singular, and the equations they describe may or may not have solutions. If there is a solution there will be infinitely many of them.

If by homogeneuos you mean Ax=0 then yes there will be an infinite number of solutions as trivially if x satisfies Ax=0 then tx is a solution for any number t, and one of the equivalent statements for determinant zero is that there is *a* non-trivial solution to Ax=0 (ie one where x is not the zero vector).

 

[M98Ranger]

First of all, welcome to the community and thank you for your question! There are no dumb questions, especially when it comes to learning and understanding new concepts.

To answer your questions, yes, if the determinant of a square matrix is not equal to zero, then the matrix has a unique solution. This is known as a non-singular matrix, which means it is invertible and has a unique solution. In other words, the columns of the matrix are linearly independent, and there are no redundant equations in the system.

On the other hand, if the determinant is equal to zero, then the matrix is singular, meaning it is not invertible and does not have a unique solution. In this case, there are either infinitely many solutions or no solutions at all, depending on the specific values in the matrix.

For a homogeneous system, if the determinant is equal to zero, then the system has infinitely many solutions, as you correctly stated. This is because the system represents a set of equations where all the constants are equal to zero, and there are multiple combinations of variables that satisfy these equations.

I hope this helps clarify your understanding of determinants and their relationship to solutions in square matrices. Keep asking questions and keep learning!