WEBVTT
00:00:00.550 --> 00:00:19.940
Use the inverse matrix to solve the matrix equation, where the three-by-three matrix with elements two, three, four; negative five, five, six; and seven, eight, nine is multiplied by the column matrix with elements ๐ฅ, ๐ฆ, and ๐ง is equal to the column matrix with elements zero, 45, and negative 45, giving your answer as an appropriate matrix.
00:00:20.770 --> 00:00:30.540
Weโre given a system of linear equations in the form ๐ด multiplied by ๐ฎ is equal to ๐ฏ, where ๐ด is a three-by-three matrix, ๐ฎ is a column matrix, and ๐ฏ is a column matrix.
00:00:31.090 --> 00:00:35.000
The column matrix ๐ฎ contains the three unknowns ๐ฅ, ๐ฆ, and ๐ง.
00:00:35.000 --> 00:00:39.270
And weโre asked to solve this system using the matrix inverse.
00:00:39.750 --> 00:00:44.870
What this means is that assuming ๐ด is nonsingular, that is, an inverse exists for ๐ด.
00:00:45.330 --> 00:00:50.760
If we find the inverse of ๐ด, we can multiply our equation through on the left by ๐ด inverse.
00:00:51.190 --> 00:00:59.890
And we can use the fact that ๐ด inverse multiplied by ๐ด is equal to ๐ด๐ด inverse, which is equal to the identity for an ๐-by-๐ nonsingular matrix.
00:01:00.290 --> 00:01:11.250
Recalling that the ๐-by-๐ identity matrix is the matrix whose elements are all zero except those on the leading diagonal which are all equal to one, which for a three-by-three matrix is as shown.
00:01:12.250 --> 00:01:17.210
On the left-hand side then, we have ๐ด inverse ๐ด, which is equal to ๐ผ, multiplied by ๐ฎ.
00:01:17.780 --> 00:01:20.230
And ๐ผ multiplied by ๐ฎ is simply ๐ฎ.
00:01:20.780 --> 00:01:26.790
And so weโve isolated ๐ฎ on the left-hand side, remembering that ๐ฎ is our column matrix with elements ๐ฅ, ๐ฆ, ๐ง.
00:01:26.790 --> 00:01:28.510
And these are our unknowns.
00:01:29.040 --> 00:01:32.330
And on our right-hand side, we have ๐ด inverse multiplied by ๐ฏ.
00:01:32.760 --> 00:01:36.240
So if we find ๐ด inverse, we can solve for ๐ฅ, ๐ฆ, and ๐ง.
00:01:36.990 --> 00:01:40.680
To find the inverse of the matrix ๐ด, weโre going to use the adjoint method.
00:01:41.300 --> 00:01:49.790
That is, for an ๐-by-๐ nonsingular matrix ๐ด, the inverse of ๐ด is given by one over the determinant of ๐ด multiplied by the adjoint of ๐ด.
00:01:50.500 --> 00:01:55.390
This means weโll have to find the determinant of the matrix ๐ด and the adjoint of matrix ๐ด.
00:01:55.940 --> 00:02:19.130
For the determinant, we can expand along the first row of our matrix ๐ด so that we have the determinant of ๐ด is equal to the element ๐ one one, which is two, multiplied by the determinant of the two-by-two matrix minor ๐ด one one minus the element ๐ one two multiplied by the determinant of its matrix minor plus the element ๐ one three, which is four, multiplied by the determinant of its matrix minor.
00:02:19.720 --> 00:02:25.880
Recalling that the matrix minor ๐ด ๐๐ is the matrix ๐ด with row ๐ and column ๐ removed.
00:02:26.410 --> 00:02:35.670
So, for example, in our case, the matrix minor๐ด one one is the two-by-two matrix resulting in the removal of row one and column one from our matrix ๐ด.
00:02:36.200 --> 00:02:41.940
Matrix minor ๐ด one one is therefore the two-by-two matrix with elements five, six, eight, and nine.
00:02:42.550 --> 00:02:50.010
Similarly, for our second term, the matrix minor ๐ด one two is the two-by-two matrix with elements negative five, six, seven, and nine.
00:02:50.440 --> 00:02:57.230
And for our third term, the matrix minor ๐ด one three is the two-by-two matrix with elements negative five, five, seven, and eight.
00:02:57.830 --> 00:03:01.890
To evaluate this, weโre going to need to work out our two-by-two determinants.
00:03:03.020 --> 00:03:10.780
And we recall that for a two-by-two matrix ๐ with elements ๐, ๐, ๐, ๐, the determinant of ๐ is equal to ๐๐ minus ๐๐.
00:03:11.240 --> 00:03:17.510
So, for example, in our first term, the determinant is five multiplied by nine minus six multiplied by eight.
00:03:18.060 --> 00:03:26.930
That is 45 minus 48, which is negative three, so that our first term evaluates to two times negative three, which is negative six.
00:03:27.330 --> 00:03:35.880
For our second term, weโll have negative three multiplied by negative five times nine minus six times seven, which is negative three times negative 87.
00:03:36.390 --> 00:03:44.760
And for our third term, we have four multiplied by negative five times eight minus five times seven, which is four times negative 75.
00:03:45.290 --> 00:03:52.310
Our determinant is then negative six plus 261 minus 300, which is negative 45.
00:03:52.840 --> 00:03:57.440
And now that we have the determinant of matrix ๐ด, we need to find its adjoint matrix.
00:03:58.110 --> 00:04:09.560
Now making some room and making a note of our determinant, we recall that the adjoint of a matrix is the transpose of the matrix of its cofactors, which for a three-by-three matrix is as shown.
00:04:10.170 --> 00:04:19.670
For element ๐ ๐๐, the cofactor ๐ ๐๐ is negative one raised to the power ๐ plus ๐ multiplied by the determinant of the matrix minor ๐ด ๐๐.
00:04:20.140 --> 00:04:25.760
And note that negative one raised to the power ๐ plus ๐ gives us the parity or sign of the cofactor.
00:04:26.240 --> 00:04:33.210
For a three-by-three matrix, thatโs positive, negative, positive; negative, positive, negative; and positive, negative, positive.
00:04:33.890 --> 00:04:38.850
Weโve actually already seen three of the cofactors when calculating the determinant of our matrix.
00:04:39.150 --> 00:04:43.420
๐ one one is the positive determinant of matrix minor ๐ด one one.
00:04:43.910 --> 00:04:51.640
And that is five multiplied by nine minus six multiplied by eight, which is 45 minus 48, which, as we saw before, is negative three.
00:04:52.070 --> 00:04:56.840
Similarly, our second cofactor ๐ one two is negative negative 87.
00:04:57.340 --> 00:04:58.870
Thatโs positive 87.
00:04:59.280 --> 00:05:02.830
And our third cofactor ๐ one three is negative 75.
00:05:03.180 --> 00:05:06.150
So now letโs write out the second two rows of cofactors.
00:05:06.590 --> 00:05:13.470
Our cofactor ๐ two one is five, ๐ two two is negative 10, and ๐ two three is five.
00:05:13.830 --> 00:05:21.640
And our final row of cofactors are ๐ three one is negative two, ๐ three two is negative 32, and ๐ three three is 25.
00:05:22.150 --> 00:05:28.360
And note that itโs very important that we get our parity correct, which is positive, negative, positive, and so on.
00:05:29.360 --> 00:05:31.650
So now we can write out our adjoint matrix.
00:05:31.680 --> 00:05:42.400
Thatโs the transpose of the matrix with elements negative three, 87, negative 75; five, negative 10, and five; negative two, negative 32, and 25.
00:05:42.950 --> 00:05:53.570
Now making some room, we can write out our adjoint matrix, which is the transpose of the matrix of cofactors, where to obtain the transpose, we interchange the rows and columns of the matrix.
00:05:53.870 --> 00:06:04.120
This means that our first row becomes our first column, our second row becomes our second column, and our third row becomes our third column.
00:06:04.660 --> 00:06:08.930
Now remember, itโs actually the inverse of our matrix ๐ด that weโre looking for at the moment.
00:06:09.360 --> 00:06:17.680
And the inverse is one over the determinant multiplied by the adjoint matrix, which in our case is one over negative 45 times our adjoint matrix.
00:06:17.990 --> 00:06:20.860
And of course we can move our negative sign to the numerator.
00:06:21.740 --> 00:06:27.520
Now remember, our original equation is ๐ฎ is equal to ๐ด inverse multiplied by ๐ฏ.
00:06:27.940 --> 00:06:31.170
And writing this out with our matrices, we have the equation as shown.
00:06:31.820 --> 00:06:36.590
And to solve this, we simply multiply out the right-hand side and use equality of matrices.
00:06:37.050 --> 00:06:44.220
Before we do this, however, we can simplify things a little if we multiply our matrix ๐ฏ by negative one over 45.
00:06:44.850 --> 00:06:52.840
This will give us zero, negative 45 over 45, which is negative one, and negative negative 45 over 45, which is positive one.
00:06:53.440 --> 00:06:57.050
This then eliminates our scalar factor of negative one over 45.
00:06:57.080 --> 00:07:03.910
So on the right, we replace negative one over 45 times ๐ฏ with the column matrix with elements zero, negative one, one.
00:07:04.180 --> 00:07:13.360
Using matrix multiplication then, we have negative three multiplied by zero plus five multiplied by negative one plus negative two multiplied by one.
00:07:13.890 --> 00:07:27.820
And similarly, for our second and third rows, we have 87 multiplied by zero plus negative 10 times negative one plus negative 32 times one and in the third row negative 75 times zero plus five times negative one plus 25 times one.
00:07:28.340 --> 00:07:32.650
These three rows then evaluate to negative seven, negative 22, and 20.
00:07:33.000 --> 00:07:41.190
An appropriate matrix for the solution of our matrix equation is therefore the column matrix with elements negative seven, negative 22, and 20.
00:07:41.860 --> 00:07:44.210
And these are our values for ๐ฅ, ๐ฆ, and ๐ง.