For any 3-3 matrix M- let -M- denote the determinant of M- Let I be the 3-3 identify matrix- Let E and F be two 3-3 matrices such that I-EF is invertible- If G - I -EF-1- then which of the following statements isare TRUE-

G = (I EF)^ 1 ⇒ G(I EF)=(I EF)G=I ⋯(1) |I FE||FGE| =|FGE FEFGE| =|FGE F(G I)E| nbsp; (From (1)) =|FGE FGE+FE| =|FE| ∴ |FE|=|I FE||FGE| (I FE) (I + FGE) ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

For any 3×3 m...

Question

For any $3 \times 3$ matrix $M$ , let $| M |$ denote the determinant of $M$ . Let $I$ be the $3 \times 3$ identify matrix. Let $E$ and $F$ be two $3 \times 3$ matrices such that $(I - E F)$ is invertible. If $G = (I - E F)^{- 1}$ , then which of the following statements is(are) TRUE?

| F E | = | I - F E | | F G E |

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(I - F E) (I + F G E) = I

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E F G = G E F

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(I - F E) (I - F G E) = I

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Solution

The correct option is C $E F G = G E F$
$G = (I - E F)^{- 1}$
$\Rightarrow G (I - E F) = (I - E F) G = I \dots (1)$

$| I - F E | | F G E |$
$= | F G E - F E F G E |$
$= | F G E - F (G - I) E |$ (From $(1)$ )
$= | F G E - F G E + F E |$
$= | F E |$
$∴ | F E | = | I - F E | | F G E |$

$(I - F E) (I + F G E)$
$= I + F G E - F E - F E F G E$
$= I + F G E - F E - F (G - I) E$ (From $(1)$ )
$= I + F G E - F E - F G E + F E$
$= I$
$∴ (I - F E) (I + F G E) = I$

From $(1),$ we have
$G (I - E F) = (I - E F) G$
$\Rightarrow G - G E F = G - E F G$
$\Rightarrow G E F = E F G$

$(I - F E) (I - F G E)$
$= I - F G E - F E + F E F G E$
$= I - F G E - F E + F (G - I) E$ (From $(1)$ )
$= I - F G E - F E + F G E - F E$
$= I - 2 F E \neq I$

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