Question

Let A,B, and C be n×n invertible matrices. Simplify the expression as much as possible. Show all work for the simplification.

(AC⁻¹)⁻¹ (AC⁻¹) (ABᵀ)ᵀ (CAᵀ)⁻¹

Answer 1

The matrix expression simplifies to BC⁻¹, by utilizing matrix properties such as the inverse and transpose.

Step-by-step explanation:

The given matrix expression can be simplified by utilizing matrix multiplication properties, such as associativity, distribution of transpose over matrix multiplication, and the inverse of a product. The expression is (AC⁻¹)⁻¹ (AC⁻¹) (ABᵀ)ᵀ (CAᵀ)⁻¹. When simplifying, we first notice that (AC⁻¹)⁻¹ is the inverse of AC⁻¹, effectively canceling out AC⁻¹ when they are multiplied together. Next, we consider that the transpose of a product of matrices is the product of the transposes in reverse order, so (ABᵀ)ᵀ = BABAᵀᵀ. Lastly, since CAᵀ is an invertible matrix, its inverse can be written as A⁻¹C⁻¹. Putting it together:

(AC⁻¹)⁻¹ (AC⁻¹) (ABᵀ)ᵀ (CAᵀ)⁻¹ = I BA A⁻¹C⁻¹

Here, I is the identity matrix. Now, using the associative property, we can group BA A⁻¹ together:

I BA A⁻¹C⁻¹ = B (AA⁻¹) C⁻¹

Since AA⁻¹ is the identity matrix, it simplifies to:

B C⁻¹

Therefore, the simplified expression is BC⁻¹.