Why Square Matrices Have Inverses: Understanding the Relationship with Determinants

a square matrix has an inverse iff

A square matrix has an inverse if and only if its determinant is non-zero

A square matrix has an inverse if and only if its determinant is non-zero. In other words, if the determinant of a square matrix is zero, then it does not have an inverse.

To understand why this is the case, let’s first define the inverse of a matrix. Given a square matrix A, the inverse of A, denoted as A^(-1), is a matrix such that when A is multiplied by A^(-1), the result is the identity matrix I.

Mathematically, A * A^(-1) = I (where I is the identity matrix).

If the determinant of A is zero, it means that the matrix is singular, which implies that there exist non-zero vectors x for which Ax = 0. In this case, if we try to find an inverse matrix A^(-1), there does not exist a matrix such that A * A^(-1) = I because multiplying A by A^(-1) would result in the zero matrix instead of the identity matrix.

On the other hand, if the determinant of A is non-zero, it means that the matrix is non-singular or invertible. This implies that for every non-zero vector x, the equation Ax = b has a unique solution (where b is any given vector). In this case, the inverse matrix A^(-1) does exist, and multiplying A by A^(-1) will indeed give us the identity matrix, as required.

To summarize, a square matrix has an inverse if and only if its determinant is non-zero.

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