### Video Transcript

If π΄ is a square matrix of order
two by two and the determinant of two π΄ is equal to 12, then the determinant of
three times π΄ transpose is equal to what. Is it option (A) 18, option (B) 24,
option (C) 27? Or is it option (D) 36?

In this question, weβre given some
information about the determinant of a two-by-two matrix π΄. Weβre told the determinant of two
π΄ is equal to 12, and we need to use this information to determine the determinant
of three times the transpose of π΄. Since weβre told that π΄ is a
two-by-two matrix, we might be tempted to start by defining π΄ to be a matrix of
four unknowns. We could then substitute our
expression for π΄ into our equation to find an expression for the determinant of π΄
and then try to use this to find an expression for the determinant of three times
the transpose of π΄. And this would work; however, it
would be very complicated. Instead, we need to notice that our
equations involve determinants of matrices.

So instead, weβll start by
simplifying by using the properties of determinants. Weβll start by simplifying the
expression the determinant of two times π΄. And to do this, weβll start by
recalling the following property. For any square matrix π΅ of order
π by π and any scalar value π, the determinant of π times π΅ is equal to π to
the πth power multiplied by the determinant of π΅. In our case, π΄ is a matrix of
order two by two. So, our value of π is two, giving
us the determinant of two π΄ is equal to two squared multiplied by the determinant
of π΄, which is, of course, four times the determinant of π΄. We can then substitute this
expression for the determinant of two π΄ into the equation weβre given in the
question. This then gives us that four times
the determinant of π΄ is equal to 12. And we can solve for the
determinant of π΄. We divide both sides of the
equation by four. This gives us the determinant of π΄
is equal to three.

However, weβre not asked to find
the determinant of π΄; weβre asked to find the determinant of three times the
transpose of π΄. To do this, letβs try simplifying
this expression by using the properties of determinants. First, we recall when we take the
transpose of a matrix, we switch the rows with the columns. So, the transpose of matrix π΄ is
also a matrix of order two by two. This means we can once again apply
the same property. π΄ transpose is a two-by-two
matrix. Therefore, the determinant of three
multiplied by the transpose of π΄ is equal to three squared multiplied by the
determinant of π΄ transpose. And we can simplify this to get
nine multiplied by the transpose of π΄, but we can simplify this expression even
further by using another one of our properties of determinants.

We recall for any square matrix π΅,
the determinant of π΅ transpose is just equal to the determinant of π΅. And we know π΄ transpose is a
square matrix, so we can replace this with the determinant of π΄ to get nine
multiplied by the determinant of π΄. And we know what the determinant of
matrix π΄ is. Itβs equal to three. Therefore, we can just substitute
three for the determinants of π΄ to get nine times three, which is equal to 27,
which we can see is given as option (C).

Therefore, weβve shown if π΄ is a
square matrix of order two by two and the determinant of two π΄ is equal to 12, then
the determinant of three times the transpose of π΄ is equal to 27.