Video Transcript
If the determinant of 𝐴 times 𝐵
is equal to 18 and the determinant of 𝐴 is equal to two, find the determinant of
𝐵.
In this question, we’re given some
information about two matrices, 𝐴 and 𝐵. We’re told the determinant of
matrix 𝐴 multiplied by matrix 𝐵 is 18 and the determinant of matrix 𝐴 is equal to
two. We need to use this information to
determine the determinant of 𝐵. To answer this question, let’s
start by recalling the property of determinants, which links the products of
matrices with the determinant of each individual matrix.
We recall if 𝐴 and 𝐵 are square
matrices of the same order, then the determinant of 𝐴 times 𝐵 is equal to the
determinant of 𝐴 multiplied by the determinant of 𝐵. We might be tempted to apply this
property directly to our question. However, there’s a problem; we’re
not told the orders of matrices 𝐴 and 𝐵. And to apply this property, we do
need the matrices 𝐴 and 𝐵 are square matrices of the same order. However, we can show that this must
be the case from the information given. First, we’re told the determinant
of matrix 𝐴 is equal to two. And we recall we can only find the
determinant of square matrices, so 𝐴 is a square matrix. Similarly, the determinant of 𝐴𝐵
is equal to 18, so 𝐴 times 𝐵 is also a square matrix.
We can use this information to find
an expression for the order of matrix 𝐵. First, since 𝐴 is a square matrix,
let’s start by saying its order is of the form 𝑛 by 𝑛. Next, we don’t know the order of
matrix 𝐵. Let’s say its order is 𝑚 by
𝑙. Then, there are two different ways
of finding an expression for the order of 𝐴 times 𝐵. First, remember, whenever we
multiply two matrices together, its order will be the number of rows of our first
matrix by the number of columns of our second matrix. So, 𝐴𝐵 must have order 𝑛 by
𝑙. However, we’ve already shown that
𝐴𝐵 is a square matrix. So, the number of rows must be
equal to the number of columns. In other words, 𝑛 is equal to
𝑙. So, we can replace 𝑙 with 𝑛.
Finally, to determine the value of
𝑚, we notice we’re allowed to multiply 𝐴 on the right by matrix 𝐵. And recall for matrix
multiplication to be well defined, the number of columns of our first matrix must be
equal to the number of rows of our second matrix. Therefore, 𝑛 must be equal to
𝑚. Therefore, we’ve shown 𝐴 is a
square matrix, 𝐵 is a square matrix, and the orders of matrices 𝐴 and 𝐵 are
equal. So, we can just apply our property
to the question. We have the determinant of 𝐴𝐵 is
equal to the determinant of 𝐴 multiplied by the determinant of 𝐵. Next, we’re told in the question
the determinant of 𝐴𝐵 is 18 and the determinant of 𝐴 is two. So, 18 is equal to two times the
determinant of 𝐵. Finally, we can divide both sides
of the equation through by two to get the determinant of 𝐵 is nine.
Therefore, we were able to show if
the determinant of 𝐴𝐵 is 18 and the determinant of 𝐴 is two, then the determinant
of 𝐵 must be equal to nine.
