Question

# A,B,C are 3 times 3 matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is det(A^2BC^{-1})? -3 -18 18 -108 36

Matrices
A,B,C are $$3 \times 3$$ matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is $$det(A^2BC^{-1})$$?
-3
-18
18
-108
36

2020-11-11
Step 1
Given A, B and C are $$3 \times 3$$ matrices.
and
$$|A|=-3$$
$$|B|=-2$$
$$|C|=6$$
Find the determinant of $$(A^2BC^{-1})$$
Step 2
since,
$$|AB|=|A| \cdot |B|$$
$$|A^n|=|A|^n$$
and
$$|A^{-1}|=\frac{1}{|A|}$$
Then,
$$|A^2BC^{-1}|=|A^2||B||C^{-1}|$$
$$=|A^2||B| \times \frac{1}{|C|}$$
Substitute all the given values,
Step 3
$$|A^2BC^{-1}|=(-3)^2(-2) \times \frac{1}{6}$$
$$=9 \times (-2) \times \frac{1}{6}$$
$$=-18 \times \frac{1}{6}$$
$$=-3$$
Hence,
$$|A^2BC^{-1}|=-3$$