Using the elementary row operation,
find the inverse of 𝐴 for the matrix 𝐴 is equal to five, three, two, one if
For a two-by-two matrix, 𝐴 is
equal to 𝑎, 𝑏, 𝑐, 𝑑, the inverse of 𝐴 is given by the formula one over the
determinant of 𝑎 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the
determinant of 𝑎 is given by 𝑎 multiplied by 𝑑 minus 𝑏 multiplied by 𝑐.
Notice that this means if the
determinant of the matrix 𝐴 is zero, the inverse cannot exist. That’s because one over the
determinant of 𝐴 would be one over zero, which we know to be undefined. Let’s substitute the values for our
matrix 𝐴 into our formula first for the determinant of 𝐴.
𝑎 multiplied by 𝑑 is five
multiplied by one, and 𝑏 multiplied by 𝑐 is three multiplied by two. Evaluating, we get five minus six,
which is negative one. Since the determinant of 𝐴 is not
zero, the inverse of 𝐴 does indeed exist. Now that we have the determinant,
we can substitute everything we know about our matrix into the formula for the
inverse of 𝐴.
That’s one divided by negative one
multiplied by one, negative three, negative two, five. Remember, we switch the values of
the five and the one, and we make three and two negative. One divided by negative one is just
We can next multiply each of the
individual elements in the matrix by negative one. And doing so, we can see that the
inverse of 𝐴 is equal to negative one, three, two, negative five.