a square matrix has an inverse iff
A square matrix has an inverse if and only if it is non-singular or invertible
A square matrix has an inverse if and only if it is non-singular or invertible. In other words, a square matrix A has an inverse if and only if its determinant is non-zero.
To understand why this is the case, let’s consider a square matrix A. To find its inverse, we need to find another matrix B such that when you multiply A and B, you get the identity matrix I.
Mathematically, we can write this as AB = BA = I.
Now, if A is invertible, then there exists a matrix B such that AB = BA = I. In this case, we say that B is the inverse of A and denote it as A^(-1).
On the other hand, if A is not invertible or singular, it means that there is no matrix B that satisfies AB = BA = I. In this case, A does not have an inverse.
The determinant of a square matrix plays a crucial role in determining if it is invertible. If the determinant of A is non-zero, i.e., det(A) ≠ 0, then A is invertible. However, if the determinant is zero, i.e., det(A) = 0, then A is not invertible.
To summarize, a square matrix has an inverse if and only if its determinant is non-zero. If the determinant is zero, the matrix is not invertible.
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