Let
\(A = \begin{pmatrix} 1+i & 1 \\ -i & 0 \end{pmatrix}\) where \(i=\sqrt{−1}.\)
Then, the number of elements in the set
\(\left\{n∈\left\{1,2,…,100\right\}:A^n=A\right\}\)
is ________.
Let
\(A = \begin{pmatrix} 1+i & 1 \\ -i & 0 \end{pmatrix}\) where \(i=\sqrt{−1}.\)
Then, the number of elements in the set
\(\left\{n∈\left\{1,2,…,100\right\}:A^n=A\right\}\)
is ________.
Correct Answer: 25
Solution and Explanation
The correct answer is 25
\(\therefore A^2 = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix} \begin{bmatrix} 1+i & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} i & 1+i \\ 1-i & -i \end{bmatrix}\)
\(A^4 = \begin{bmatrix} i & 1+i \\ 1-i & -i \end{bmatrix} \begin{bmatrix} i & 1+i \\ 1-i & -i \end{bmatrix} = l\)
So A5 = A, A9 = A and so on.
Clearly n = 1, 5, 9, ….., 97
Thereore , number of values of n = 25
Top Questions on Matrices
- Which of the following statements is not correct?
- 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:
- If \( B = \begin{bmatrix} 1 & 3 \\ 2 & \alpha \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( |A| = 2 \), then the value of \( \alpha \) is:
- If a matrix \( A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \) satisfies \( A^6 = kA' \), then the value of \( k \) is:
Questions Asked in JEE Main exam
- Find the value of the integral: \[ \int_0^e \log_e x \, dx \]
- JEE Main - 2025
- Integral Calculus
- The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
- JEE Main - 2025
- Binomial theorem
- A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:
- JEE Main - 2025
- Quantum Mechanics
- Which of the following quantity has the same dimensions as \( \sqrt{\frac{\mu_0}{\epsilon_0}} \)?
- JEE Main - 2025
- Units, Dimensions and Measurements
- Number of solutions in the interval \( [-2\pi, 2\pi] \) for the equation: \[ 2\sqrt{2} \cos^2 \theta + (2 - \sqrt{6}) \cos \theta - \sqrt{3} = 0 \]
- JEE Main - 2025
- Trigonometry
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.