00:02
In this question, we are given the value of the determinant of the matrix, let's call this matrix b, we are asked to calculate the determinant of the matrix 3a inverse.
00:13
First of all, recall the determinant of the original matrix is equal to the determinant of the transpose matrix.
00:21
So the determinant of b equals to the determinant of b transpose and b transpose equals to a, b, c in the first column, pqr in the second column, and uvdb in the last column.
00:37
To get the transpose matrix, you simply need to write rows as columns and vice versa.
00:47
Now, in the matrix a, recall another property of determinants, that if you factor out a number from a column, so let's say you have determinant of a, determinant of some matrix, let's call it r.
01:12
Then if you factor out some number from some row of the matrix r, then the whole determinant gets multiplied by that number.
01:27
So, for example, if you take the determinant of the matrix, a time so on a, and some other entries, it doesn't matter.
01:35
And if you want to factor out a, then you get the whole determinant multiplied by a times determinant once in the first row and everything else is same.
01:47
In our case, if we factor out 7 from the first column of the matrix a, we are going to get 7, then we will factor out 4 from the second column, and then we will factor out negative 1 from the last column, and then the remaining matrix will be a, b, c, u, v, w, and p, q, and r.
02:23
That's very similar to the matrix b, but we need to interchange the second, sorry, to be transposed, but we need to interchange the second and the last columns.
02:37
When you interchange columns in the determinant, you need to, the determinant gets multiplied by negative 1.
02:44
So we are going to get negative 28 multiplied by negative 1 times the determinant of the matrix b...