Enter two numbers to check if the first number is divisible by the second number: Check
Problem:
Let and
be
and
, respectively, such that
Determine with proof the matrix
.
Solution:
It is easy to check that (why?) and
(details).
Since the first row and the second row of is linearly independent then
and since
is not invertible (why?) then
. So, we can conclude that
.
We obtain the determinant of is
Thus, is not invertible.
Another argument to show that is not invertible is if we assume it is invertible then we can multiply the equation
both side by
and we obtain
which is contradiction since .
Since then
Therefore, since the size of is
then the rank of
must be
. Thus,
is an invertible matrix.
Now, we obtain that
.
Finally, since , multiply both side with
we obtain
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