Find the matrix 'X'.

From part (i):

A = [[2,1], [1,2]]

Let:

X = [[x], [y]]

Given:

AX = B

[[2,1], [1,2]] [[x], [y]] = [[4], [5]]

Multiplying:

[[2x + y], [x + 2y]] = [[4], [5]]

Therefore:

2x + y = 4 ...(1)

x + 2y = 5 ...(2)

Multiply equation (1) by 2:

4x + 2y = 8 ...(3)

Subtract equation (2) from equation (3):

3x = 3

x = 1

Substitute in equation (1):

2(1) + y = 4

y = 2

Therefore:

X = [[1], [2]]

Answer:

X = [[1],[2]].

The matrix equation is converted into two simultaneous linear equations and solved step by step.