Let
\[
R = \begin{pmatrix}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z
\end{pmatrix}
\text{ be a non-zero } 3 \times 3 \text{ matrix, where}
\]
\[
x = \sin \theta, \quad y = \sin \left( \theta + \frac{2\pi}{3} \right), \quad z = \sin \left( \theta + \frac{4\pi}{3} \right)
\]
and \( \theta \neq 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). For a square matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:
- \(\text{Trace}(R) = 0\)
- If \(\text{trace}(\text{adj}(\text{adj}(R))) = 0\), then \( R \) has exactly one non-zero entry.
Which of the following is true?
Let \[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \text{ be a non-zero } 3 \times 3 \text{ matrix, where} \]
\[ x = \sin \theta, \quad y = \sin \left( \theta + \frac{2\pi}{3} \right), \quad z = \sin \left( \theta + \frac{4\pi}{3} \right) \]
and \( \theta \neq 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). For a square matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:
- \(\text{Trace}(R) = 0\)
- If \(\text{trace}(\text{adj}(\text{adj}(R))) = 0\), then \( R \) has exactly one non-zero entry.
Which of the following is true?
- Only(I) is true
- Only (II) is true
- Neither (I) nor (II) is true
- Both (I) and (II) are true
The Correct Option is A
Approach Solution - 1
This problem requires us to analyze two statements regarding a non-zero \( 3 \times 3 \) diagonal matrix \( R \). The diagonal entries of \( R \) are given as trigonometric functions of an angle \( \theta \).
Concept Used:
1. Trace of a Matrix: The trace of a square matrix is the sum of its diagonal elements, denoted as \( \text{trace}(M) = \sum_{i} M_{ii} \).
2. Trigonometric Identities: We will use the sum-of-angles formula for sine:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
3. Adjoint of a Matrix: For an \( n \times n \) matrix \( M \), the second adjoint (adjoint of the adjoint) is given by the formula:
\[ \text{adj}(\text{adj}(M)) = (\det M)^{n-2} M \]
4. Determinant of a Diagonal Matrix: The determinant of a diagonal matrix is the product of its diagonal entries.
Step-by-Step Solution:
Analysis of Statement 1: Trace(R) = 0
Step 1: Write down the expression for the trace of matrix \( R \).
The matrix is \( R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \). The trace is the sum of its diagonal entries:
\[ \text{trace}(R) = x + y + z \]
Step 2: Substitute the given trigonometric definitions for \( x, y, \) and \( z \).
\[ \text{trace}(R) = \sin \theta + \sin \left( \theta + \frac{2\pi}{3} \right) + \sin \left( \theta + \frac{4\pi}{3} \right) \]
Step 3: Expand the terms using the sine sum-of-angles formula.
We know the values:
- \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2}, \quad \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \)
- \( \cos \left( \frac{4\pi}{3} \right) = -\frac{1}{2}, \quad \sin \left( \frac{4\pi}{3} \right) = -\frac{\sqrt{3}}{2} \)
Expanding \( y \) and \( z \):
\[ y = \sin \left( \theta + \frac{2\pi}{3} \right) = \sin \theta \cos \left( \frac{2\pi}{3} \right) + \cos \theta \sin \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta \] \[ z = \sin \left( \theta + \frac{4\pi}{3} \right) = \sin \theta \cos \left( \frac{4\pi}{3} \right) + \cos \theta \sin \left( \frac{4\pi}{3} \right) = -\frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta \]
Step 4: Sum the terms to find the trace.
\[ \text{trace}(R) = \sin \theta + \left( -\frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta \right) + \left( -\frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta \right) \] \[ \text{trace}(R) = \left( \sin \theta - \frac{1}{2} \sin \theta - \frac{1}{2} \sin \theta \right) + \left( \frac{\sqrt{3}}{2} \cos \theta - \frac{\sqrt{3}}{2} \cos \theta \right) \] \[ \text{trace}(R) = (0) \sin \theta + (0) \cos \theta = 0 \]
This result holds for any value of \( \theta \). Therefore, the statement "Trace(R) = 0" is true.
Analysis of Statement 2: If \( \text{trace}(\text{adj}(\text{adj}(R))) = 0 \), then \( R \) has exactly one non-zero entry.
Step 1: Find the expression for \( \text{trace}(\text{adj}(\text{adj}(R))) \).
For a \( 3 \times 3 \) matrix \( R \), the second adjoint is \( \text{adj}(\text{adj}(R)) = (\det R)^{3-2} R = (\det R) R \).
Since \( R \) is a diagonal matrix, its determinant is the product of its diagonal elements:
\[ \det R = xyz \]
So, the second adjoint is:
\[ \text{adj}(\text{adj}(R)) = (xyz) R = (xyz) \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} = \begin{pmatrix} x^2yz & 0 & 0 \\ 0 & xy^2z & 0 \\ 0 & 0 & xyz^2 \end{pmatrix} \]
The trace of this matrix is:
\[ \text{trace}(\text{adj}(\text{adj}(R))) = x^2yz + xy^2z + xyz^2 = xyz(x + y + z) \]
Step 2: Evaluate the condition \( \text{trace}(\text{adj}(\text{adj}(R))) = 0 \).
From our analysis of Statement 1, we know that \( x + y + z = 0 \). Substituting this into the trace expression:
\[ \text{trace}(\text{adj}(\text{adj}(R))) = xyz(0) = 0 \]
This shows that the hypothesis of Statement 2, \( \text{trace}(\text{adj}(\text{adj}(R))) = 0 \), is always true for any \( \theta \).
Step 3: Check the conclusion of Statement 2: "R has exactly one non-zero entry."
The statement is an implication: If A is true, then B must be true. We have shown A is always true. Now we must check if B is always true. Let's test a value for \( \theta \) that is allowed by the problem conditions, e.g., \( \theta = \frac{\pi}{6} \).
\[ x = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] \[ y = \sin\left(\frac{\pi}{6} + \frac{2\pi}{3}\right) = \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \] \[ z = \sin\left(\frac{\pi}{6} + \frac{4\pi}{3}\right) = \sin\left(\frac{3\pi}{2}\right) = -1 \]
For this value of \( \theta \), the matrix \( R \) has diagonal entries \( \frac{1}{2}, \frac{1}{2}, -1 \). All three entries are non-zero. This provides a counterexample to the conclusion that \( R \) must have exactly one non-zero entry.
Since the hypothesis of the implication is always true but the conclusion is not always true, the implication itself is false.
Final Computation & Result:
Based on the analysis:
- Statement 1, "Trace(R) = 0", is true.
- Statement 2, "If trace(adj(adj(R))) = 0, then R has exactly one non-zero entry", is false.
Therefore, only the first statement is true.
Approach Solution -2
Calculate the trace of \( R \): Since \( x + y + z = \sin \theta + \sin \left( \theta + \frac{2\pi}{3} \right) + \sin \left( \theta + \frac{4\pi}{3} \right) = 0 \), we have:
\[ \text{trace}(R) = x + y + z = 0. \]
Thus, statement (I) is true.
Examine statement (II): \(\text{adj}(R) = \begin{pmatrix} yz & 0 & 0 \\ 0 & xz & 0 \\ 0 & 0 & xy \end{pmatrix}\). Therefore,
\[ \text{adj}(\text{adj}(R)) = \begin{pmatrix} x^2yz & 0 & 0 \\ 0 & xy^2z & 0 \\ 0 & 0 & xyz^2 \end{pmatrix}. \]
The trace of \(\text{adj}(\text{adj}(R))\) is \( xyz(x + y + z) = 0 \), even if \( R \) has more than one non-zero entry.
Thus, statement (II) is false.
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Matrices Questions
- Let \( A \) be a \( 2 \times 2 \) real matrix and \( I \) be the identity matrix of order 2. If the roots of the equation \[ |A - xI| = 0 \] be \( -1 \) and \( 3 \), then the sum of the diagonal elements of the matrix \( A^2 \) is .....
- Consider the matrix \( f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \).
Given below are two statements:
Statement I: \( f(-x) \) is the inverse of the matrix \( f(x) \).
Statement II: \( f(x) f(y) = f(x + y) \).
In the light of the above statements, choose the correct answer from the options given below:" - Let \( A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} \), \( B = [B_1, B_2, B_3] \), where \( B_1, B_2, B_3 \) are column matrices, and \( AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \), \( AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \), \( AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \).
If \(\alpha = |B|\) and \(\beta\) is the sum of all the diagonal elements of B, then \(\alpha^3 + \beta^3\) is equal to _____. - If \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), then \( \det (X) \) is equal to:
- If the system of linear equations
\( x - 2y + z = -4 \)
\( 2x + \alpha y + 3z = 5 \)
\( 3x - y + \beta z = 3 \)
has infinitely many solutions, then \( 12\alpha + 13\beta \) is equal to
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
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.