GATE EC 2016 Set 1 — Question 11
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Engineering Mathematics → Linear Algebra → Matrix Algebra
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Question
Let , (where denotes the identity matrix) and , and . Then, for any natural number , equals:
Correct answer
(C) M^(4k + 3)
Solution
To find the expression for the inverse matrix , we start with the given matrix equation:Since , the matrix is invertible. Multiplying both sides of the equation by from the left (or right), we get:Thus, the inverse of is given by:We are also given that . This implies that for any integer :Now, we can express in terms of by multiplying it by (since multiplying by does not change the matrix):Using the laws of exponents for matrices:To match the given options, we can rewrite by utilizing the relation again:Let us verify this for any natural number :This matches Option C.
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