Question

Matrix
A
=
[
2
−
3
6
1
]
and Matrix
B
=
[
2
−
1
4
−
2
]
.
Stephanie concluded that
A
B
=
B
A
.
Is her conclusion valid?






Her conclusion is valid because matrix multiplication is associative.


Her conclusion is valid because
A
B
=
[
4
3
24
−
2
]
and
B
A
=
[
4
3
24
−
2
]
.


Her conclusion is not valid because matrix multiplication is not associative.


Her conclusion is not valid because
A
B
=
[
−
8
4
16
−
8
]
and
B
A
=
[
−
2
−
7
−
4
−
14
]
.

Answer 1

Her conclusion is not valid.

A = [[2, -3],[6, 1]], B = [[2, -1],[4, -2]].

AB = [[2*2 + (-3)*4, 2*(-1) + (-3)*(-2)],
[6*2 + 1*4, 6*(-1) + 1*(-2)]]
= [[-8, 4],[16, -8]].

BA = [[2*2 + (-1)*6, 2*(-3) + (-1)*1],
[4*2 + (-2)*6, 4*(-3) + (-2)*1]]
= [[-2, -7],[-4, -14]].

Since AB ≠ BA, the conclusion is false (matrix multiplication is not generally commutative). This matches the fourth choice.