In conclusion, a matrix A is invertible if and only if its determinant, (det(A)), equals ±1, establishing a direct relationship between invertibility and the determinant's magnitude.
To show that a matrix A is invertible if and only if det(A) = ±1, we can use the fact that the determinant of a product of matrices is the product of their determinants and the fact that the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
First, assume that A is invertible. Then there exists a matrix A^-1 such that A * A^-1 = A^-1 * A = I, where I is the identity matrix. Taking determinants on both sides, we get det(A * A^-1) = det(I) = 1. Since det(A * A^-1) = det(A) * det(A^-1), we have det(A) * det(A^-1) = 1. Since A is invertible, det(A^-1) ≠ 0, so we can divide both sides by det(A^-1), giving us det(A) = ±1.
Conversely, if det(A) = ±1, then det(A^-1) = 1/det(A) = ±1. This implies that A^-1 exists, making A invertible.