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 \(α=\frac{π}{3}\)
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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.