Math 108B Selected Homework Solutions

(b) We know that T is invertible if and only if its matrix is invertible with respect to any basis; we also know that a matrix is invertible if and only if its determinant is zero. If A is a matrix representing T in some basis, then T is invertible if and only if A is invertible if and only if det(T ) = det(A) 6= 0. (c) If T has matrix A under some basis, then T−1 has matrix A−1 under the same basis. Thus det(T−1) = det(A−1) = det(A)−1 = det(T )−1.