Final answer:
When a matrix A is invertible, its determinant (det A) is non-zero. This can be proven by showing that the determinant of a product of matrices is equal to the product of their determinants. If A is invertible, then matrix BA is also invertible.
Step-by-step explanation:
When we are given that a matrix A is invertible, it means that its determinant (det A) is non-zero. To prove this, we can use the fact that the determinant of a product of matrices is equal to the product of their determinants. Since A is invertible, it means that there exists another matrix B such that AB = BA = I (the identity matrix). Let's compute the determinant of AB and BA:
- Determinant of AB: det(AB) = det(A)det(B) = det(B)det(A) = det(BA)
- Determinant of BA: det(BA) = det(B)det(A)
From the above two equations, we can conclude that det(AB) = det(BA). Since A is invertible, its determinant is non-zero, which means det(A) ≠ 0. Therefore, we have det(AB) = det(BA) ≠ 0, which implies that det(BA) is also non-zero. Therefore, matrix BA is also invertible.