Matrices & Determinants
Determinants
MJAT None
Grade 12
Question:
For any $3 \times 3$ matrix $\mathbf{M}$, let $|\mathbf{M}|$ denote the determinant of $\mathbf{M}$. Let $\mathbf{I}$ be the $3 \times 3$ identity matrix. Let $\mathbf{E}$ and $\mathbf{F}$ be two $3 \times 3$ matrices such that $(\mathbf{I} - \mathbf{E}\mathbf{F})$ is invertible. If $\mathbf{G} = (\mathbf{I} - \mathbf{E}\mathbf{F})^{-1}$, then which of the following statements is (are) TRUE ?
A) $|\mathbf{F}\mathbf{E}| = |\mathbf{I} - \mathbf{F}\mathbf{E}||\mathbf{F}\mathbf{G}\mathbf{E}|$
B) $(\mathbf{I} - \mathbf{F}\mathbf{E})(\mathbf{I} + \mathbf{F}\mathbf{G}\mathbf{E}) = \mathbf{I}$
Step-by-Step Solution
Key Concept: The inverse of a matrix can be found using the formula involving the determinant and adjugate matrix, which is a key concept in linear algebra.
$$mathbf{G} = (mathbf{I} - mathbf{E}mathbf{F})^{-1}$$ can be found using the formula for the inverse of a matrix. Since $(mathbf{I} - mathbf{E}mathbf{F})$ is invertible, we can use the identity $$mathbf{A}^{-1} = rac{1}{|mathbf{A}|} ext{adj}(mathbf{A})$$ where $ ext{adj}(mathbf{A})$ is the adjugate matrix of $mathbf{A}$. Therefore, $$mathbf{G} = rac{1}{|mathbf{I} - mathbf{E}mathbf{F}|} ext{adj}(mathbf{I} - mathbf{E}mathbf{F})$$
Correct Answer: A, B