Step 1: Understanding the Concept:
We are dealing with matrix identities. Note that \( A^{-1}A = I \) (the identity matrix). The notation \( B' \) refers to the transpose \( B^T \).
Step 2: Key Formula or Approach:
1. \( (PQ)^T = Q^T P^T \).
2. \( (P^{-1})^T = (P^T)^{-1} \).
3. Given: \( AA^T = I \) and \( B = A^{-1}A^T \).
Step 3: Detailed Explanation:
We need to find \( BB^T \).
\[ B = A^{-1}A^T \]
Then \( B^T = (A^{-1}A^T)^T = (A^T)^T (A^{-1})^T = A(A^T)^{-1} \).
Now, calculate the product \( BB^T \):
\[ BB^T = (A^{-1}A^T)(A(A^T)^{-1}) \]
From the given condition \( AA^T = I \), we know \( A^T = A^{-1} \) and \( A = (A^T)^{-1} \).
Substitute \( A^T = A^{-1} \) and \( A = (A^T)^{-1} \) into the expression:
\[ BB^T = (A^{-1}A^{-1})(A \cdot A) \]
Alternatively, grouping differently:
\[ BB^T = A^{-1}(A^T A)(A^T)^{-1} \]
Since \( AA^T = I \), the matrix \( A \) is orthogonal, which implies \( A^T A = I \) as well.
\[ BB^T = A^{-1}(I)(A^T)^{-1} = A^{-1}(A^T)^{-1} = (A^T A)^{-1} = I^{-1} = I \]
In the context of the options, \( I \) is represented as 1.
Step 4: Final Answer:
The product \( BB' \) is equal to 1 (Identity matrix).