Simplifying the Matrix Product (PQ)^–1P

The question asks us to find the simplified form of the matrix expression $$(PQ)^{-1}P$$. To solve this, we need to use the properties of matrix inverses and multiplication.

Recall the property for the inverse of a product of matrices:

• For any invertible matrices A and B of the same size, $$(AB)^{-1} = B^{-1}A^{-1}$$.

Applying this property to the term $$(PQ)^{-1}$$, we get:

$$(PQ)^{-1} = Q^{-1}P^{-1}$$

Now, substitute this back into the original expression $$(PQ)^{-1}P$$:

$$ (PQ)^{-1}P = (Q^{-1}P^{-1})P $$

Matrix multiplication is associative, meaning we can group the matrices differently:

$$ (Q^{-1}P^{-1})P = Q^{-1}(P^{-1}P) $$

Next, we use the property that a matrix multiplied by its inverse results in the identity matrix, denoted by $$I$$:

• For any invertible matrix A, $$A^{-1}A = AA^{-1} = I$$.

Using this property for matrix P:

$$P^{-1}P = I$$

Substitute $$I$$ back into our expression:

$$ Q^{-1}(P^{-1}P) = Q^{-1}I $$

Finally, multiplying any matrix by the identity matrix $$I$$ results in the original matrix:

• For any matrix A, $$AI = IA = A$$.

Applying this property:

$$ Q^{-1}I = Q^{-1} $$

Therefore, the simplified product of matrices $$(PQ)^{-1}P$$ is $$Q^{-1}$$.

Let's review the steps:

1. Apply the inverse of a product rule: $$(PQ)^{-1} = Q^{-1}P^{-1}$$.
2. Substitute this into the expression: $$(PQ)^{-1}P = Q^{-1}P^{-1}P$$.
3. Use associativity: $$Q^{-1}(P^{-1}P)$$.
4. Apply the inverse property $$P^{-1}P = I$$: $$Q^{-1}I$$.
5. Apply the identity property $$Q^{-1}I = Q^{-1}$$.

The final result is $$Q^{-1}$$.