If \(A=\begin{bmatrix}cosα& -sinα\\ sinα& cosα\end{bmatrix}\),thenA+A=I,if the value of \(α\) is
- \(\frac{π}{6}\)
\(\frac{π}{3}\)
- π
- \(\frac{3π}{2}\)
The Correct Option is B
Solution and Explanation
The correct option is(B): \(\frac{π}{3}\)
\(A=\begin{bmatrix}cosα& -sinα\\ sinα& cosα\end{bmatrix}\)
\(A'=\begin{bmatrix}cosα& sinα\\ -sinα& cosα\end{bmatrix}\)
Now A+A=I
\(\begin{bmatrix}cosα& -sinα\\ sinα& cosα\end{bmatrix}+\begin{bmatrix}cosα& sinα\\ -sinα& cosα\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
\(=>\begin{bmatrix}2cosα& 0\\ 0& 2cosα\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
Comparing the corresponding elements of the two matrices, we have:
2cosα=1
=>\(cos a=\frac{1}{2}=cos\frac{\pi}{3}\)
Therefore \(α=\frac{π}{3}\)
Top Questions on Matrices
Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $A^2 = A^T$, then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to
- If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:
Let $ A $ be a $ 3 \times 3 $ matrix such that $ | \text{adj} (\text{adj} A) | = 81.
$ If $ S = \left\{ n \in \mathbb{Z}: \left| \text{adj} (\text{adj} A) \right|^{\frac{(n - 1)^2}{2}} = |A|^{(3n^2 - 5n - 4)} \right\}, $ then the value of $ \sum_{n \in S} |A| (n^2 + n) $ is:- Let $ A $ be a matrix of order $ 3 \times 3 $ and $ |A| = 5 $. If $ |2 \, \text{adj}(3A \, \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} $ then $ \alpha + \beta + \gamma $ 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:
Questions Asked in CBSE CLASS XII exam
- A 1 cm segment of a wire lying along the x-axis carries a current of 0.5 A along the \( +x \)-direction. A magnetic field \( \mathbf{B} = (0.4 \, \text{mT}) \hat{j} + (0.6 \, \text{mT}) \hat{k} \) is switched on in the region. The force acting on the segment is:
- CBSE CLASS XII - 2025
- Magnetic Effects of Current and Magnetism
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is:
- CBSE CLASS XII - 2025
- Coordination Compounds
- Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
- CBSE CLASS XII - 2025
- Evaluation of Definite Integrals by Substitution
- Two conductors A and B of the same material have their lengths in the ratio 1:2 and radii in the ratio 2:3. If they are connected in parallel across a battery, the ratio \( \frac{v_A}{v_B} \) of the drift velocities of electrons in them will be:
- CBSE CLASS XII - 2025
- Current electricity
- Arun, Bashir and Joseph were partners in a firm sharing profits and losses in the ratio of 5 : 3 : 2. They admitted Daksh as a new partner who acquired his share entirely from Arun. If Arun sacrificed \( \frac{1}{5} \)th from his share to Daksh, Daksh's share in the profits of the firm will be :
- CBSE CLASS XII - 2025
- Partnership Accounts
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.