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

If 𝐴 is a square matrix of order 2 Γ— 2 and |2𝐴| = 12, then |3𝐴^(𝑇)| = οΌΏ.

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Video Transcript

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

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

However, we’re not asked to find the determinant of 𝐴; we’re asked to find the determinant of three times the transpose of 𝐴. To do this, let’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 𝐴 is also a matrix of order two by two. This means we can once again apply the same property. 𝐴 transpose is a two-by-two matrix. Therefore, the determinant of three multiplied by the transpose of 𝐴 is equal to three squared multiplied by the determinant of 𝐴 transpose. And we can simplify this to get nine multiplied by the transpose of 𝐴, but we can simplify this expression even further by using another one of our properties of determinants.

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

Therefore, we’ve shown if 𝐴 is a square matrix of order two by two and the determinant of two 𝐴 is equal to 12, then the determinant of three times the transpose of 𝐴 is equal to 27.

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