a square matrix has an inverse iff
the determinant is nonzero
A square matrix has an inverse if its determinant is not zero.
In other words, if a square matrix A is invertible, then there exists a matrix B such that the product of A and B is the identity matrix. This can be expressed mathematically as:
A * B = B * A = I
where I is the identity matrix.
Now, consider the determinant of both sides of this equation:
det(A * B) = det(B * A) = det(I)
Using the property of determinants that det(X * Y) = det(X) * det(Y), we can rewrite this as:
det(A) * det(B) = det(B) * det(A) = det(I)
Since the determinant of the identity matrix is 1, we can simplify this equation to:
det(A) * det(B) = 1
Therefore, if det(A) is not equal to zero, then there exists a matrix B such that A * B = B * A = I, and A is invertible. However, if det(A) is equal to zero, then there does not exist an inverse because the product of A with any matrix B will not equal the identity matrix.
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