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
- The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$, is
- If \[ X=\begin{bmatrix}x\\y\\z\end{bmatrix} \] is a solution of the system of equations $AX=B$, where \[ \text{adj }A= \begin{bmatrix} 4 & 2 & 2\\ -5 & 0 & 5\\ 1 & -2 & 3 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix}4\\0\\2\end{bmatrix}, \] then $|x+y+z|$ is equal to
Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that
- For the matrices \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} \), if \( (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} \), then among the following which one is true?}
- Let the relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) be given by \[ R = \{(x, y) : 4y = 5x - 3,\; x, y \in M\}. \] Then the minimum number of elements required to be added in \( R \), in order to make the relation symmetric, is equal to
Questions Asked in JEE Advanced exam
- Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?
- JEE Advanced - 2025
- Fundamental Theorem of Calculus
- A linear octasaccharide (molar mass = 1024 g mol$^{-1}$) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is 58.26 % (w/w) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is _____.
Use: Molar mass (in g mol$^{-1}$): ribose = 150, 2-deoxyribose = 134, glucose = 180; Atomic mass (in amu): H = 1, O = 16- JEE Advanced - 2025
- Biomolecules
Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Conic sections
- Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from 10 mg g$^{-1}$ and 16 mg g$^{-1}$ aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be 4 mg g$^{-1}$ and 10 mg g$^{-1}$, respectively. At this temperature, the concentration (in mg g$^{-1}$) of adsorbed phenol from 20 mg g$^{-1}$ aqueous solution of phenol will be ____. Use: $\log_{10} 2 = 0.3$
- JEE Advanced - 2025
- Adsorption
- At 300 K, an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height (h) of the solution (density = 1.00 g cm$^{-3}$) where h is equal to 2.00 cm. If the concentration of the dilute solution of the macromolecule is 2.00 g dm$^{-3}$, the molar mass of the macromolecule is calculated to be $X \times 10^{4}$ g mol$^{-1}$. The value of $X$ is ____. Use: Universal gas constant (R) = 8.3 J K$^{-1}$ mol$^{-1}$ and acceleration due to gravity (g) = 10 m s$^{-2}\}$
- JEE Advanced - 2025
- Colligative Properties
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.