
Let be an integer
and is the identity matrix of order 3, then following of which is correct
A. and
B. for any positive integer
C. is not invertible
D. for a positive integer
Answer
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Hint: Here we will first find the square of the given matrix and then we will find its cube. Then we will see that it is forming a certain pattern. We will follow the same pattern to find the matrix raised to given power. We will simplify the matrix using trigonometric identities and find the correct answer.
Complete step-by-step answer:
The given matrix is .
Let and we will substitute this value here.
Now, we will find the product of matrices .
We can write as .
Now, we will multiply these matrices using the rule of multiplication of matrices. Therefore, we get
On further simplifying the terms, we get
Now using the trigonometric identities and in the above matrix, we get.
Similarly, we will find the product of matrices .
We can write as .
Now, we will multiply these matrices using the rule of multiplication of matrices.
Now using the trigonometric identities and in the above matrix, we get
We can see that it is following a certain pattern as shown below:
Now, we will substitute the value in the above matrix. Therefore, we get
On further multiplying the terms, we get
Now, substituting and in the above matrix, we get
We know that
Therefore, we have
Also,
Hence, the correct option is option A.
Note: To solve this question, we need to know the meaning or definition of the trigonometric identities. Trigonometric identities are defined as the equalities which involve the trigonometric functions. They are always true for every value of the occurring variables for which both sides of the equality are defined. We need to remember that all the trigonometric identities are periodic in nature. They repeat their values after a certain interval. These intervals are a multiple of .
Complete step-by-step answer:
The given matrix is
Let
Now, we will find the product of matrices
We can write
Now, we will multiply these matrices using the rule of multiplication of matrices. Therefore, we get
On further simplifying the terms, we get
Now using the trigonometric identities
Similarly, we will find the product of matrices
We can write
Now, we will multiply these matrices using the rule of multiplication of matrices.
Now using the trigonometric identities
We can see that it is following a certain pattern as shown below:
Now, we will substitute the value
On further multiplying the terms, we get
Now, substituting
We know that
Therefore, we have
Also,
Hence, the correct option is option A.
Note: To solve this question, we need to know the meaning or definition of the trigonometric identities. Trigonometric identities are defined as the equalities which involve the trigonometric functions. They are always true for every value of the occurring variables for which both sides of the equality are defined. We need to remember that all the trigonometric identities are periodic in nature. They repeat their values after a certain interval. These intervals are a multiple of
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