Question

If `A = ((a, 0, 0),(0, a, 0),(0, 0, a))`, then Aⁿ equal to ______.

Options

• `((a^n, 0, 0),(0, a^n, 0),(0, 0, a^n))`

• `((a, 0, 0),(0, a^n, 0),(0, 0, a))`

• `((a^n, 0, 0),(0, a, 0),(0, 0, a^n))`

• `((na, 0, 0),(0, na, 0),(0, 0, a^n))`

Answer 1

If `A = ((a, 0, 0),(0, a, 0),(0, 0, a))`, then Aⁿ equal to `underlinebb(((a^n, 0, 0),(0, a^n, 0),(0, 0, a^n)))`.

Explanation:

Given: `A = ((a, 0, 0),(0, a, 0),(0, 0, a))`

The matrix A is a scalar multiple of the identity matrix, which can be written as A = aI, where `I = ((1, 0, 0),(0, 1, 0),(0, 0, 1))` is the identity matrix.

To find Aⁿ, we can raise this expression to the power of n:

Aⁿ = (aI)ⁿ

Using the property of matrix multiplication that (kI)ⁿ = kⁿIⁿ = kⁿI, where k is a scalar:

Aⁿ = aⁿI

`A^n = a^n((1, 0, 0),(0, 1, 0),(0, 0, 1))`

`A^n = ((a^n, 0, 0),(0, a^n, 0),(0, 0, a^n))`