# Linear Algebraic Problems (Math Problem Sample)

Determine if matrix a is idempotent. A square matrix A is idempotent if A2=A.

Prove that if A is an n×n matrix that is idempotent and invertible.

A square matrix A is idempotent if A2=A.

Prove that if A is an n×n matrix that is idempotent and invertible, then A is the identity.A square matrix A is idempotent if A2=A.

Prove that if A is an n×n matrix that is idempotent and invertible, then A is the identity.

How do i prove this?

How do i prove this?

How do i prove this?

Find whetheR matrix A iS invertible.

Find eigen value corresponding to matrix A

Linear Algebra Problem

* Since a is idempotent, we have

A2 = A

(I – A)2 = I2 + A2 – 2IA

(I – A)2 = I + A -A

(I – A)2 = I – A

Implying that I – A is idempotent

* Since A is invertible, A-1 exist

A2 = A

Multiply both sides by A-1

A-1(AA) = A-1A

(A-1A)A = A-1A

IA = I

A = I

* Let λ be the eigen value of A corresponding to the vector

Then, Av = λv

A2v = λv

A(λv) = λv

A(Av) = λv