# Question Video: Finding the Determinant of the Scalar Multiple of a Matrix Mathematics

If π΄ is a square matrix of order 2 Γ 2 and |2π΄| = 12, then |3π΄^(π)| = οΌΏ.

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### 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, weβre given some information about the determinant of a two-by-two matrix π΄. Weβre 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 weβre 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, weβll start by simplifying by using the properties of determinants. Weβll start by simplifying the expression the determinant of two times π΄. And to do this, weβll 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 πth 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 weβre 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, weβre not asked to find the determinant of π΄; weβre asked to find the determinant of three times the transpose of π΄. To do this, letβs 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. Itβs 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, weβve 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.