Question

Use MATLAB to write your codes Consider a matrix A with block matrices as follows: A = {A-11 A_12 0 A-22] It can be shown that the inverse of A can be calculated by inverse of submatrices if A11, and A22 are squared matrices: A^-1 = [A_11 A_12 0 A_22] = [A6-1 _11 -A^-1 _11 A_12 A^-1 _22 0 A^-1 _22 Now consider a Matrix A with following submatrices: A11 = identity matrix A22 = identity matrix A12 = [12 3: 4 5-1] Write a code that check the identity shown above: steps are: a) Takes 4 sub matrices: A11, A22, A21, A12 b) Given the information about the matrix A, construct A: a. What is the size of A? (create a zero matrix with that size) c) create a variable LHS which uses matlab built-in function to take the inverse (calculate LHS) d) create a variable RHS which reconstruct the inverse given in the right hand side of the identity e) show that they are equal

Answer 1

a)

MATLAB CODE

A11 = eye(2,2)
A12 = [1 2 3;4 5 -1]
A22 = eye(3,3)
A21 = zeros(3,2)

OUTPUT

Command Window >A11 eye (2,2) A12[1 2 3:4 5 -1] A22eye (3,3) A21 - zeros (3,2) A11 0 A12 5 -1 A22 A21 fx. >>

b)

MATLAB CODE

A = [A11 A12;A21 A22]

size(A)

OUTPUT

Command Window >> A = [A11 A12:A21 A22] size (A) 5 -1 ans - fx. >>

c)

MATLAB CODE

LHS=pinv(A)

OUTPUT

omman >> LHS=pinv (A) LHS = 0.0000 -0.0000 -0.0000 0.0000 1.0000 -0.0000-1.0000 -2.0000 -3.0000 1.0000 0.0000 0.0000 1.0000 1.0000-4.0000-5.0000 1.0000 -0.0000 1.0000 0.0000 -0.0000 -0.0000 0 0.0000 -0.0000 fx. >>

d)

MATLAB CODE

a = pinv(A11)
d = pinv(A22)
b = -a*A12*d

RHS = [a b;A21 d]

OUTPUT

Command Window >> a = pǐnv (A11) d = pins(A22) b--a A12*d a= d= RHS 01-23 1-4-5

e)

MATLAB CODE

if isequal(int64(LHS),RHS)
disp 'Success'
end

OUTPUT