Number 3 only please thanks 2 Show that the projection matr

Number 3 only please! :) thanks!

2. Show that the projection matrix P X (X\'X) X\' is symmetric and idempotent. 3. Show that the matrix M E I Px is symmetric and idempotent

Solution

M_x = I - P_x

The matrix A is symmetric if A^T = A

Now M_x^T = (I-P_x)^T = I^T-P_x^T = I-P_x (Because I being identity matrix is symmteric and also P_x is symmetric from part2)

Hence M_x^T = M_x => M_x is symmetric

A matrix A is idempotent if and only if, A² = A

Now M_x² = (I-P_x)² = (I-P_x)(I-P_x) = I.I-IP_x-P_xI+P_x.P_x = I² - P_x-P_x+P_x² (Because I being identity , so I.P_x = P_xI = P_x)

=> M_x² = I - 2P_x+P_x (I being identity is idempotent and P_xis idempotent because of part 2 so P_x²=P_x)

=> M_x² = I-P_x = M_x => M_x is idempotent