# If A and B are 3×3 invertible matrices, such that det(A)=2, det(B) =-2. Then det (ABA^T) = ???

Question
Matrices
If A and B are 3×3 invertible matrices, such that det(A)=2, det(B) =-2. Then det (ABA^T) = ???

2020-11-08
Step 1
To solve this problem, we use the standard result of matrix
Ans: We know that
$$det(A \cdot B)= det(A) \cdot det(B)$$
and
$$det(A)=det(A^T)$$\)
Step 2
Ans : Given that $$det(A)=2, det(B)=-2$$
$$det(ABA^T)=det(A) \cdot det(BA^T) \ \ \ \left\{det(A \cdot B)=det(A) \cdot det(B)\right\}$$
$$=det(A) \cdot det(B) \cdot det(A^T) \ \ \ \left\{det(A)=det(A^T)\right\}$$
$$=det(A) \cdot det(B) \cdot det(A)$$
$$=2 \cdot (-2) \cdot 2$$
$$det(ABA^T)=-8$$

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