Question

If A is a square matrix such that A² = A, then (A − I)³ − A is equal to ______.

Options

• I

• −I

• A

• A²

Answer 1

If A is a square matrix such that A² = A, then (A − I)³ − A is equal to −I.

Explanation:

Use the standard binomial expansion formula for (A − I)³

(A − I)³ = A³ − 3A²I + 3AI² − I³

Substituting I² = I and I³ = I, we get:

(A − I)³ = A³ − 3A² + 3A − I

Given:

A² = A

A³ = A . A²

= A . A

= A²

= A

Substituting these back into the expanded expression:

Substitute A³ = A and A² = A into the expanded form of (A − I)³:

(A − I)³ = A − 3(A) + 3A − I

(A − I)³ = A − 3A + 3A − I

(A − I)³ = A − I

Solve the final expression:

(A − I)³ − A = (A − I) − A

= A − I − A

= −I