Find the multiplicative inverse of
𝐴 is equal to the matrix negative four, eight, negative 12, 24, if possible.
As we have no way of dividing
matrices, finding the inverse is an important tool that helps us solve problems.
In this question, we’re given the
two-by-two matrix negative four, eight, negative 12, 24. And we recall that for any
two-by-two matrix 𝐴 equal to 𝑎, 𝑏, 𝑐, 𝑑, then the inverse of 𝐴 is equal to one
over the determinant of 𝐴 multiplied by the two-by-two matrix 𝑑, negative 𝑏,
negative 𝑐, 𝑎, where the determinant is calculated by finding the product of
elements 𝑎 and 𝑑 and subtracting the product of 𝑏 and 𝑐. This means that if the determinant
is zero, the inverse does not exist, since we cannot divide by zero.
It therefore makes sense to find
the determinant of our matrix first. Since matrix 𝐴 is equal to
negative four, eight, negative 12, 24, then the determinant of this matrix is equal
to negative four multiplied by 24 minus eight multiplied by negative 12. This is equal to negative 96 minus
negative 96, which simplifies to negative 96 plus 96 and in turn is equal to
Since the determinant is equal to
zero, there is no inverse of our matrix 𝐴. And we can therefore conclude that
matrix 𝐴 has no multiplicative inverse.