WEBVTT
00:00:01.880 --> 00:00:18.680
Given the two-by-two matrices π΄, which is equal to eight, negative three, one, negative two, and π΅, which is equal to eight, negative three, one, negative two, is π΄π΅ equal to π΅π΄?
00:00:20.440 --> 00:00:23.600
So weβve been given two two-by-two matrices.
00:00:24.160 --> 00:00:28.360
And weβre asked whether π΄π΅ is equal to π΅π΄.
00:00:29.120 --> 00:00:37.760
So weβre being asked whether we get the same result when we multiply the two matrices together in different orders.
00:00:38.640 --> 00:00:49.480
In general, we know that matrix multiplication is not commutative, which means that we get a different result if we multiply the matrices together in a different order.
00:00:50.480 --> 00:00:58.360
In fact, unless the two matrices are each square matrices of the same order, it wonβt even be possible to find both products.
00:00:59.560 --> 00:01:03.920
Matrix multiplication can be commutative under special circumstances.
00:01:04.560 --> 00:01:13.200
For example, if both matrices are diagonal matrices of the same order, meaning that they are square matrices.
00:01:14.040 --> 00:01:18.840
And all of the elements that arenβt on the leading diagonal are equal to zero.
00:01:19.400 --> 00:01:24.960
Matrix multiplication will also be commutative if one matrix is the identity matrix.
00:01:25.800 --> 00:01:30.200
And the other is a square matrix of the same order.
00:01:31.200 --> 00:01:34.960
Now, neither of these conditions apply here.
00:01:35.480 --> 00:01:41.440
But if we look at our two matrices π΄ and π΅, we see that they do have another relationship.
00:01:42.120 --> 00:01:45.160
π΄ and π΅ are each two-by-two matrices.
00:01:45.840 --> 00:01:47.520
So they have the same order.
00:01:48.520 --> 00:01:55.040
But more importantly, each of their elements in corresponding positions are the same.
00:01:56.160 --> 00:02:00.640
In this case then, matrix π΄ is entirely equal to matrix π΅.
00:02:01.320 --> 00:02:05.560
For this reason, it doesnβt matter in which order we multiply these two matrices together.
00:02:06.160 --> 00:02:10.000
The product π΄π΅ will be equal to the product π΅π΄.
00:02:10.720 --> 00:02:14.440
And in fact, theyβll both be equal to π΄ squared or indeed π΅ squared.
00:02:15.320 --> 00:02:25.080
We could of course confirm this by multiplying the matrices together longhand and checking that the two products do indeed give the same result.
00:02:25.800 --> 00:02:26.960
But there is no need.
00:02:27.400 --> 00:02:39.400
Once we spotted that the two matrices are identical, we know that their product will be the same regardless of which way around we find it.
00:02:40.240 --> 00:02:49.520
So we can conclude that, for these two particular matrices π΄ and π΅, but not in general, π΄π΅ is equal to π΅π΄.