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) \( EFG = GEF \)
Top Questions on Matrices
- If A and B are invertible matrices of order $3 \times 3$ such that $\text{det} = 4$ and $\text{det}([AB]^{-1}) = \frac{1}{20}$, then $\text{det}$ is equal to:
- If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:
- For all \( n \in \mathbb{N} \), if \( 1^3 + 2^3 + 3^3 + \cdots + n^3>x \), then a value of \( x \) among the following is:
- If A and B are two matrices such that AB is an identity matrix and the order of matrix B is \(3 \times 4\), then the order of matrix A is:
- If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:
Questions Asked in JEE Advanced exam
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- binomial expansion formula
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
- JEE Advanced - 2025
- Thermodynamics
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
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.