CHAPTER 04.10: UNARY MATRIX
OPERATIONS: Theorems on determinants Part 2 of 4 In
this segment we’ll talk about some more theorems on determinants. So let A be
a n by n matrix. So again, we’re talking about
square matrices because determinants can only be found for a square matrix.
So let A be a n by n matrix. Then is a row is proportional
to another row, then the determinant of A is equal to 0. SO if you find out
that one row is proportional to another row or a multiple of another row,
other way of saying the same thing, then the determinant of the whole matrix
is 0. And this is true for columns also. Then if a column is proportional to another
column – another column, in that case also the determinant of A will be equal
to 0. So if one row is proportional to another row or if one column is proportional
to another column, then the determinant of A is equal to 0. Let’s go and look
at this through an example. Let’s suppose somebody says hey A is 3, 6, -9,
3.6, 7.2, -10.8, 5, 15, 19. So in this case what you
are going to find out here is that you got 3, 6, -9, 3.6, 7.2, -10.8, 5, 15,
19; that’s the A matrix. You can clearly see that this particular row here is
1.2 times this particular row here. Because 3 times 1.2 is equal to 3.6; 6
times 1.2 is equal to 7.2; and -9 times 1.2 is equal to -10.8. What we are
finding out is that since this role is proportional to this row or this row
is a multiple of this row, then the determinant of A would be equal to 0. Let’s
look at another example in which we may find that the columns are proportional.
So let’s suppose somebody takes this particular
matrix here and asks you to find the determinant, but if you are able to
recognize that hey, this particular column is two times this particular
column. So this column is proportional to this column right here by multiple
of two. As you an see that
3 times 2 is equal to 6; 6 times 2 is equal to 12; -9 times 2 is equal to -18.
It has to be the same number of course. Then the determinant of the matrix is
equal to 0. So if a row is proportional to another row or if a column is
proportional to another column, the determinant of the whole matrix is equal
to 0. And that’s the end of this segment. |