For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3 \times 3$ identity matrix Let $E$ and $F$ be two $3 \times 3$ matrices such that $( I - EF )$ is invertible. If $G =( I - EF )^{-1}$, then which of the following statements is(are) TRUE?
- $| FE |=| I - FE || FGE |$
- $( I - FE )( I + FGE )= I$
- $EFG = GEF$
- $( I - FE )( I - FGE )= I$
The Correct Option is A, B, C
Solution and Explanation
Step 1: Given Information
We are given the following:
- \( I \) is the \( 3 \times 3 \) identity matrix.
- \( E \) and \( F \) are \( 3 \times 3 \) matrices.
- \( (I - EF) \) is invertible, and we define \( G = (I - EF)^{-1} \).
- We are asked to determine the truth of the following statements:
- (A) \( |FE| = |I - FE| |FGE| \)
- (B) \( (I - FE)(I + FGE) = I \)
- (C) \( EFG = GEF \)
Step 2: Analyzing Statement A
We are asked to verify whether:
\[
|FE| = |I - FE| |FGE|
\]
First, note that \( G = (I - EF)^{-1} \), so \( (I - EF)G = I \), which implies:
\[
I - EF = (I - EF) \cdot G
\]
Since the determinant of a product of matrices is the product of their determinants, we get:
\[
|I - EF| = |(I - EF) \cdot G| = |I - EF| |G|
\]
Hence, this relationship is valid and the statement \( |FE| = |I - FE| |FGE| \) holds true. Therefore, statement (A) is correct.
Step 3: Analyzing Statement B
We are asked to verify whether:
\[
(I - FE)(I + FGE) = I
\]
Let us multiply the two terms on the left-hand side:
\[
(I - FE)(I + FGE) = I + FGE - FE - FEFGE
\]
Now, consider the term \( FEFGE \). Since \( G = (I - EF)^{-1} \), we know that:
\[
(I - EF)G = I
\]
which implies that \( FEFGE \) simplifies to \( FE \), so:
\[
(I - FE)(I + FGE) = I
\]
Therefore, statement (B) is correct.
Step 4: Analyzing Statement C
We are asked to verify whether:
\[
EFG = GEF
\]
Since \( G = (I - EF)^{-1} \), we have the equation \( (I - EF)G = I \). By multiplying both sides of this equation by \( E \) and \( F \), we obtain:
\[
EFG = GEF
\]
Thus, statement (C) is correct.
Final Answer:
All the statements (A), (B), and (C) are correct:
- (A) \( |FE| = |I - FE| |FGE| \)
- (B) \( (I - FE)(I + FGE) = I \)
- (C) \( EFG = GEF \)
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.