Question

Suppose A, B, and C are invertible nxn matrices. Show that ABC is also invertible by introducing a matrix D such that (ABC)D = I and D(ABC) = I. It is assumed that A, B, and C are invertible matrices. What does this mean?

Α. Α⁻¹ B⁻¹ and C⁻¹ exist
B. A⁻¹ B⁻¹ and C⁻¹ are all equal to the identity matrix.
C A⁻¹ B⁻¹ and C⁻¹ are all not equal to the identity matrix.
D. A⁻¹ B⁻¹ and C⁻¹ all have determinants equal to zero.

Answer 1

To prove that the matrix product ABC is invertible, we must introduce a matrix D such that (ABC)D = I and D(ABC) = I. By taking D to be C^-1B^-1A^-1, we satisfy these conditions, demonstrating that ABC has an inverse and thus is invertible.

Step-by-step explanation:

To show that the matrix product ABC is invertible, we introduce a matrix D such that (ABC)D = I and D(ABC) = I, where I is the identity matrix. Since matrices A, B, and C are invertible, it means that A-1, B-1, and C-1 exist (Option A). The inverse of a matrix, when multiplied by the original matrix, yields the identity matrix, which is a property ensuring the existence of an invertible matrix. Therefore, if we take D to be C-1B-1A-1, we will have:

• (ABC)(C-1B-1A-1) = A(B(C(C-1)))(B-1A-1) = A(BI)(B-1A-1) = A(BB-1A-1) = AIA-1 = AA-1 = I
• (C-1B-1A-1)(ABC) = C-1(B-1(A-1(AB))C) = C-1(B-1(IB)C) = C-1((B-1B)C) = C-1(IC) = C-1C = I

Thus, D serves as an inverse to ABC, proving it is invertible.