Inverse of a Product Matrix?

📌 CONCEPT: The relationship between a matrix and its inverse helps in finding the inverse of a product matrix, given that the product matrix is equal to the product of its individual inverses in reverse order.

📐 RULE / FORMULA: To find the inverse of a product matrix ABC, we use the formula (ABC)^-1 = C^-1 * B^-1 * A^-1.

💡 WORKED EXAMPLE: Suppose A, B, and C are matrices such that ABC is the identity matrix, and D is any invertible matrix. We need to find (ABC * D)^-1. Using the formula, (ABC * D)^-1 = (D * ABC)^-1 = (D^-1 * B^-1 * C^-1) * (ABC)^-1 = D^-1 * B^-1 * C^-1.

⚠️ COMMON MISTAKE: Students often get confused between the order of the matrices when finding the inverse of a product matrix. They may write (ABC)^-1 = B^-1 * C^-1 * A^-1, which is incorrect. Therefore, it's essential to remember the correct formula and apply it carefully.