MAT1503 Assignment 2 solutions 2026

0793226427




MAT1503 Assignment 2 solutions 2026

UNISA

, 0793226427


MAT1503 Assignment 02 — Full Solutions



Question 1

Given: 𝐴3 − 3𝐴2 + 3𝐴 − 𝐼 = 0, where 𝐴is an 𝑛 × 𝑛matrix.



Part (a): Prove that A is invertible

To prove 𝐴is invertible, we need to show det⁡(𝐴) ≠ 0, i.e., find 𝐴−1explicitly, i.e.,
express 𝐼as a product of 𝐴and some matrix.

Rearrange the given equation:

𝐴3 − 3𝐴2 + 3𝐴 − 𝐼 = 0
𝐴3 − 3𝐴2 + 3𝐴 = 𝐼



Factor out 𝐴on the left side:

𝐴(𝐴2 − 3𝐴 + 3𝐼) = 𝐼



This shows that:

𝐴 ⋅ (𝐴2 − 3𝐴 + 3𝐼) = 𝐼



By the definition of invertibility, a matrix 𝐴is invertible if and only if there exists a
matrix 𝐵such that 𝐴𝐵 = 𝐼.

Here, 𝐵 = 𝐴2 − 3𝐴 + 3𝐼satisfies 𝐴𝐵 = 𝐼.

Therefore, 𝐴is invertible. ■



Part (b): Expression for 𝑨−𝟏

From Part (a):

𝐴(𝐴2 − 3𝐴 + 3𝐼) = 𝐼