Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If G = (I - EF)<sup>-1</sup>, then which of the following statements is (are) TRUE?
(A) |FE| = |I - FE||FGE|
(B) |I - FE|(I + FGE) = I
(C) EFG = GEF
(D) (I - FE)(I - FGE) = I
Step-by-Step Solution
Key Concept: Use the property of determinants |I - AB| = |I - BA| and the identity (I - AB)^-1 A = A(I - BA)^-1.
Given G = (I - EF)<sup>-1</sup>, we have G(I - EF) = I, so G - GEF = I, which implies G = I + GEF. Also (I - EF)G = I, so G - EFG = I, which implies G = I + EFG. Thus EFG = GEF. For (A), |I - FE| = |I - EF| = 1/|G|. Also |I - FGE| = |I - F(I - EF)<sup>-1</sup>E|. Using the identity |I - AB| = |I - BA|, we can show the relations hold.
Correct Answer: 1, 2, 3