MTH202 Worksheet 3¶
Problem 1¶
Find a REF of the matrix $$ A = \left[ \begin{array}{rrrr} -2 & 3 & 1 & 4 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 1 & 0 \\ 2 & 1 & 2 & -2 \\ -2 & 2 & 0 & -3 \end{array} \right] $$
In [1]:
import numpy as np
In [2]:
A = np.array([[-2, 3, 1, 4],
[0, 1, -2, -1],
[0, 0, 1, 0],
[2, 1, 2, -2],
[-2, 2, 0, -3]])
In [3]:
A[3,:] += A[0,:]
A[4,:] -= A[0,:]
A
Out[3]:
array([[-2, 3, 1, 4],
[ 0, 1, -2, -1],
[ 0, 0, 1, 0],
[ 0, 4, 3, 2],
[ 0, -1, -1, -7]])
In [4]:
A[3,:] -= 4*A[1,:]
A[4,:] += A[1,:]
A
Out[4]:
array([[-2, 3, 1, 4],
[ 0, 1, -2, -1],
[ 0, 0, 1, 0],
[ 0, 0, 11, 6],
[ 0, 0, -3, -8]])
In [5]:
A[3,:] -= 11*A[2,:]
A[4,:] += 3*A[2,:]
A
Out[5]:
array([[-2, 3, 1, 4],
[ 0, 1, -2, -1],
[ 0, 0, 1, 0],
[ 0, 0, 0, 6],
[ 0, 0, 0, -8]])
In [6]:
A[4,:] = A[4,:] + (8/6)*A[3,:] # This full code fixes the datatype conflict.
A # <-- REF
Out[6]:
array([[-2, 3, 1, 4],
[ 0, 1, -2, -1],
[ 0, 0, 1, 0],
[ 0, 0, 0, 6],
[ 0, 0, 0, 0]])
Problem 2¶
Find the RREF matrix of $$ B = \left[ \begin{array}{rrrr} 11 & 2 & 0 & -3 \\ 8 & -6 & 0 & -5 \\ 2 & 0 & 1 & 0 \end{array} \right] $$
In [7]:
B = np.array([[11, 2, 0, -3],
[8, -6, 0, -5],
[2, 0, 1, 0.]]) # Use 0. to force the datatype to be "float".
In [8]:
B[0,:] /= 11
B
Out[8]:
array([[ 1. , 0.18181818, 0. , -0.27272727],
[ 8. , -6. , 0. , -5. ],
[ 2. , 0. , 1. , 0. ]])
In [9]:
B[1,:] -= 8*B[0,:]
B[2,:] -= 2*B[0,:]
B
Out[9]:
array([[ 1. , 0.18181818, 0. , -0.27272727],
[ 0. , -7.45454545, 0. , -2.81818182],
[ 0. , -0.36363636, 1. , 0.54545455]])
In [10]:
B[1,:] /= B[1,1]
B
Out[10]:
array([[ 1. , 0.18181818, 0. , -0.27272727],
[-0. , 1. , -0. , 0.37804878],
[ 0. , -0.36363636, 1. , 0.54545455]])
In [11]:
B[0,:] -= B[0,1]*B[1,:]
B[2,:] -= B[2,1]*B[1,:]
B # <-- RREF
Out[11]:
array([[ 1. , 0. , 0. , -0.34146341],
[-0. , 1. , -0. , 0.37804878],
[ 0. , 0. , 1. , 0.68292683]])