If A = [[1, 0], [0, 1]] is the identity matrix, what is A^n for any integer n?

If A = [[1, 0], [0, 1]] is the identity matrix, what is A^n for any integer n?

Correct Answer: A^n = [[1, 0], [0, 1]]

Step 1: Understand what the identity matrix is. The identity matrix A = [[1, 0], [0, 1]] is a special matrix that, when multiplied by any other matrix of the same size, leaves that matrix unchanged.
Step 2: Recognize that raising a matrix to a power means multiplying the matrix by itself that many times. For example, A^2 means A multiplied by A.
Step 3: Calculate A^1. Since A is the identity matrix, A^1 = A = [[1, 0], [0, 1]].
Step 4: Calculate A^2. A^2 = A * A = [[1, 0], [0, 1]] * [[1, 0], [0, 1]] = [[1*1 + 0*0, 1*0 + 0*1], [0*1 + 1*0, 0*0 + 1*1]] = [[1, 0], [0, 1]] = A.
Step 5: Notice that A^2 = A. This pattern continues for any integer n. A^3 = A * A^2 = A * A = A, and so on.
Step 6: Conclude that for any integer n, A^n = A = [[1, 0], [0, 1]].

Identity Matrix – The identity matrix is a square matrix that does not change any vector when multiplied by it.
Matrix Exponentiation – Raising a matrix to a power involves multiplying the matrix by itself a specified number of times.