a square matrix has an inverse iff
the determinant is nonzero
A square matrix has an inverse if and only if it is a non-singular matrix. That is, the determinant of the matrix is non-zero. In other words, a square matrix is invertible if and only if its determinant is nonzero. If a matrix is singular, that is, the determinant of the matrix is zero, then it does not have an inverse.
The inverse of a square matrix A is denoted by A^-1, and is defined as a matrix such that the product of A and A^-1 is equal to the identity matrix I. That is,
A * A^-1 = A^-1 * A = I
where I is the identity matrix. The inverse of a matrix is unique provided that the matrix is nonsingular. If a matrix is singular, that is, the determinant of the matrix is zero, then it does not have an inverse and is said to be non-invertible.
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