Consider the determinant of the
two-by-two matrix 𝑥, 𝑦, 𝑧, 𝑤 is equal to six. Find the value of the determinant
of the two-by-two matrix 𝑥 minus 10𝑦, 𝑦, 𝑧 minus 10𝑤, 𝑤.
In this question, we’re given the
determinant of a two-by-two matrix and asked to determine the determinant of a
different two-by-two matrix. And both of these matrices contain
four unknowns, so we can’t evaluate these directly. We might be tempted to expand the
determinant in our first matrix and then expand the determinant in our second matrix
and try to rewrite this expression in terms of our first determinant. However, there’s a much simpler
method involving the properties of determinants.
To do this, we need to notice
something interesting about the second matrix we’re given. In the first entry of the first
column, we’re subtracting 10𝑦, and in the second entry of the second column, we’re
subtracting 10𝑤. This is a scalar multiple of the
second column of our first matrix. In fact, we’re subtracting this
directly from the first column of this matrix. In other words, to generate the
second matrix, we’re subtracting 10 lots of the second column from the first column
of our first matrix. And we can recall adding and
subtracting scalar multiples of one row or column to a different row or column will
not affect the determinant. Therefore, the determinant of the
second matrix will be equal to the determinant of the first matrix, which we’re told
is equal to six.