Homework answers / question archive / Suppose that we have matrix X, which is n*(q+1) with rank q+1, and matrix Z, which is n*(p+1) with rank p+1, where p > q

Suppose that we have matrix X, which is n*(q+1) with rank q+1, and matrix Z, which is n*(p+1) with rank p+1, where p > q

Math

Share With

---

Suppose that we have matrix X, which is n*(q+1) with rank q+1, and matrix Z, which is n*(p+1) with rank p+1, where p > q. Then, we calculate the matrix B = [X^TX – X^TZ(Z^TZ)^-1Z^TX]^-1.

Find the dimension and rank of B.

Let I be an n*n identity matrix and J be an n*n matrix of 1s. Let J = 1/nJ. Let Z = I – J.

1. Prove whether J is idemponent.

2. Prove whether Z is idemponent.

3. Prove whether Z is symmetric.

4. Prove whether Z is skew-symmetric.

5. Calculate tr(Z).

An idempotent matrix is one which, when multiplied by itself, yields itself. So matrix A is idempotent if A² = AA = A.

1) Given the following system of linear equations,

-w + 3x - 6y - 3z = 0

w - x + y + 4z = -11

5w – 5x - y - 2z = -1

2w + 3x - 3y = -3

(a) Evaluate whether the system of equations has a unique solution or not. Write a paragraph explaining your reasoning.

(b) If in (a) you stated that the system of equations has a unique solution, find it, You get extra 5 points if you use Gauss-Jordan elimination.

2) Given the following system of linear equations,

x + y – 3z = -1

y - z = 0

-x + 2y = 1

(a) Evaluate whether the system of equations has a unique solution or not. Write a paragraph explaining your reasoning.

(b) If in (a) you stated that the system of equations has a unique solution, find it using Gauss-Jordan elimination.

The following figure is a probability density function (p.d.f.) of the random variable X, with s.d. 2.3.

1. What is the dotted blue line illustrating in the figure?

2. For the point (x, f(x)) identified in the figure, f(x) gives us the probability of observing the value x for random variable x.

• True
• False

3. The point x in the plot is equal to 7.5. What is the purple area illustrating in the plot?²

4. Using R, calculate the area of the purple polygon in the figure.

5. Up to which value of x we have a cumulative probability of 0.34? Use R to solve this question.

Show that the function,

e(µ) = In (II^N_i=11/Ö2p e^-1/2(xi-µ)2)

is equal to,

e(m) = -In(2p)/2 N-1/2S^N_i=1(x_i - m)²

Where e is a mathematical constant, m is a constant, and x is a variable indexed by i = 1, ××× , n. For each step, mention the property you are applying or a brief note on what you are doing.