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
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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.