Square matrices have special properties that set them apart from other matrices. A square matrix has the same number of rows and columns. Singular matrices are unique and cannot be multiplied by any other matrix to get the identity matrix. Non-singular matrices are invertible, and because of this property they can be used in other calculations in linear algebra such as singular value decompositions. The first step in many linear algebra problems is determining whether you are working with a singular or non-singular matrix. (See References 1,3)
Find the determinant of the matrix. If and only if the matrix has a determinant of zero, the matrix is singular. Non-singular matrices have non-zero determinants.
Find the inverse for the matrix. If the matrix has an inverse, then the matrix multiplied by its inverse will give you the identity matrix. The identity matrix is a square matrix with the same dimensions as the original matrix with ones on the diagonal and zeroes elsewhere. If you can find an inverse for the matrix, the matrix is non-singular.
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Verify that the matrix meets all other conditions for the invertible matrix theorem to prove that the matrix is non-singular. For an "n by n" square matrix, the matrix should have a non-zero determinant, the rank of the matrix should equal "n," the matrix should have linearly independent columns and the transpose of the matrix should also be invertible.