00:01
Let's determine the adjugate of this matrix a.
00:04
By definition, we see that adjugate of matrix a.
00:14
This equals the transpose of the co -factor matrix of a.
00:34
So here we have to determine the co -factor matrix for the given matrix.
00:39
Let's see how to determine the co -factor matrix.
00:42
So basically we have to determine the co -factor of each of the elements in this matrix a.
00:58
And then we have to write down in matrix form.
01:01
And to determine the co -factor of each elements, we use this below formula, that is c -i -j, and this equals negative 1, raised to the power of i plus j times the minor of i -plus -j.
01:16
So this c -i -j represents the co -factor of the element.
01:21
In i -th row this i represents the row and j represents the column and this equals negative raised to the power of i plus j times the minor of the i j element that is minor of i -thro and jth column element so we'll be tracing this formula to determine the co -factor of each of the element i'm going to copy the matrix a here and this equals 1 -114.
01:51
312 114 312 in the second row and then 0 2 3 2 3 in the third row first we will determine the co -factor of this element so when we use this formula this will be co -factor of the position of this element is first rule and the first column then this will become i will become 1 and j will also become 1 and this equals negative 1 raised to the power of i plus j that will be 1 plus 1.
02:30
Multiply this with the minor of m1 1 because i and j are 1 1.
02:36
So this equals negative 1 raise to the power of 1 plus 1 and that is 2 times the minor.
02:44
We determine the minor by deleting the row and column of this element.
02:50
So we have to delete this row as well as this column.
02:53
And then the determinant that is obtained is the minor.
02:58
So here we have to define the determinant of 1, 2 and 2 3.
03:04
So this equals negative 1 raised to the even power and that is positive.
03:10
So basically we have to calculate the determinant.
03:12
So that equals we take the product of this diagonal that is 3 minus 3 times 1 is 3.
03:18
And then negative 2 times 2 is 4.
03:21
And so this equals negative 1.
03:27
Okay, so let's continue.
03:30
I'm going to write down this c11 here and this equals negative 1.
03:36
We remove this for some space so that we'll find the co -factor of the other elements.
03:44
Just removing this.
03:48
Now we are going to determine the co -factor of this element.
03:51
And for that, we say that it is a co -factor of first true and second column, and this equals negative 1, raise to the power of 1 plus 2, multiply this with the minor m1.
04:07
So this gives, this is 2 plus 1 is 3.
04:12
So i'm just changing this to 3.
04:15
Negative 1 raised to the odd power, and that will be negative.
04:18
So here i put negative 1.
04:20
Multiply this with m1 .2.
04:23
And for m1 .2, we have to delete this row and then this column, which means we get the determinant 3203.
04:32
Let's find the determinant of this one.
04:34
So here we have, we already have a negative.
04:37
So we put a negative.
04:39
And then product of the diagonal, 3 times 3 is 9 minus 0.
04:44
So here we get, this is 9 minus 0 is 9.
04:48
Then we have negative.
04:49
So this is 9.
04:50
So this is 9.
04:50
So this is.
04:50
Is negative 9.
04:52
So we determine that c1 .2 equals negative 9.
04:57
And i write out here c12 and this equals negative 9.
05:03
Now i'm going to find the other co -factor elements without using that formula.
05:08
And for that, i'm going to use this sign reference to determine the signs of the positions of this co -factor element.
05:17
So here this will be plus, then negative, then plus, then plus, then.
05:21
We have plus in the middle this is two negatives and plus in the corners and then we have a negative so we'll be using this sign reference to understand the signs of each of the co -factor elements now we have to calculate the co -factor of this element so for that the sign is positive and then we have to delete that through and as well as this column when we do that we are going to get let's see this is c13 and this equals.
05:55
It's basically the determinant 3 -1 and 0 -2.
05:59
So we have to evaluate this determinant.
06:01
This is 3 times 2 is 6 minus 1 times 0 is 0.
06:06
So here we get 6...