(0201 LC)
1 answer:
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Answer:
The matrix is invertible and its inverse is
.
Step-by-step explanation:
Since the definition of the inverse matrix states that the inverse of matrix is a matrix B such that:
we have to assume the form of such matrix. In our case we have the matrix and so, the constant
must be somehow eliminated from the equation. The most logical way to do so is to include
in the inverse. If we choose matrix B to be
, we will have this:
and
.
We can form the matrix B like this because we know from the text of the problem that the inverse matrix of A exists and that c is a nonzero number.
<u><em>Here is another way to solve this using the formula of the inverse matrix</em></u>
Since we know that the matrix is invertible, it follows that its determinant is different from zero. Using the formula for the inverse matrix:
we will assume the form of an inverse matrix of . We need to obtain the formula for the inverse of
, so we first need to find
. Since the matrix
is obtained from matrix
by multiplying every term with
, while calculating determinant we have a constant
that can be extracted from every column (or row) in front. Therefore, we have that
.
On the other hand, consists of minors of the matrix
. Therefore, when we extract the constant in front of such (
) determinants, we have
in each column (row). Including all this into the formula we have that:
.