Understanding Matrix Inverses: The Importance of Non-Zero Determinants for Square Matrices

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.

To understand this statement, let’s consider a square matrix A.

If A has an inverse matrix B, then the product of A and B will result in the identity matrix I, i.e., AB = BA = I. In this case, we say that A is invertible and B is the inverse of A.

Now, let’s consider the determinant of A, denoted as det(A). The determinant of a matrix is a scalar value that can be computed based on the values within the matrix. The determinant function is defined for square matrices only.

If det(A) = 0, then the matrix A is called singular, and it does not have an inverse. This is because the existence of an inverse requires the ability to “cancel out” certain terms in the multiplication process, which can only happen if the determinant is non-zero.

On the other hand, if det(A) ≠ 0, then the matrix A is called non-singular, and it has an inverse. In this case, the inverse matrix B can be computed as (1/det(A)) times the adjoint of A, where the adjoint is obtained by finding the transpose of the matrix of cofactors.

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

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