Final answer:
To prove that the product of two invertible matrices, a and b, is also invertible, one can demonstrate the existence of an inverse by showing that (ab)(b⁻¹a⁻¹) and (b⁻¹a⁻¹)(ab) both equal the identity matrix, thus establishing that (ab)⁻¹ = b⁻¹a⁻¹.
Step-by-step explanation:
The question involves proving that if a and b are invertible matrices of size n × n, then the product ab is also invertible and that the inverse of ab is given by (ab)⁻¹ = b⁻¹a⁻¹. This can be demonstrated using properties of matrix multiplication and inverses. We know that an invertible matrix has a unique inverse such that when it is multiplied by its inverse, the result is the identity matrix.
To show ab is invertible, we can show that there exists a matrix x such that abx =xba = I, where I is the identity matrix of the same size. Since a and b are invertible, we can take x to be b⁻¹a⁻¹ and perform the multiplication:
- ab(b⁻¹a⁻¹) = a(bb⁻¹)a⁻¹ = aIa⁻¹ = aa⁻¹ = I
- (b⁻¹a⁻¹)ab = b⁻¹(a⁻¹a)b = b⁻¹Ib = b⁻¹b = I
Since we have both abx = I and xab = I, we can conclude that x is indeed the inverse of ab, and therefore, (ab)⁻¹ = b⁻¹a⁻¹. This completes the proof.