If 𝐴 is a matrix of order two by two such that the determinant of 𝐴 is equal to three, find the determinant of three 𝐴.
In this question, we’re given a matrix 𝐴, which is a square matrix. It’s of order two by two. And we’re also told the determinant of 𝐴 is equal to three. We need to use all of this information to determine the value of the determinant of three 𝐴.
We might be tempted to answer this question by writing our matrix 𝐴 out as a two-by-two matrix with unknown values. We can then find an expression for the determinant of 𝐴 and set this equal to three and then find an expression for the determinant of three times 𝐴. And this would work.
However, it’s much easier to do this by using the properties of determinants. We just need to notice we’re asked to find the value of the determinant of a scalar multiplied by a square matrix. We can then recall the following property of the determinant. If 𝑀 is a square matrix of order 𝑛 by 𝑛 and 𝑘 is any scalar value, then the determinant of 𝑘 times 𝑀 is equal to 𝑘 to the 𝑛th power multiplied by the determinant of 𝑀. In other words, we can take scalar multiplication outside of our calculation of the determinant. We just need to raise this value of 𝑘 to the exponent of 𝑛, where 𝑛 is the order of our square matrix.
Since the matrix 𝐴 we’re given in the question is a square matrix, we can apply this property to evaluate the determinant of three 𝐴. 𝐴 is a matrix of order two by two. So when we take the factor of three outside our calculation of the determinant, we need to raise this to the power of two. We get three squared times the determinant of 𝐴. Three squared is equal to nine. And we’re told in the question the determinant of 𝐴 is equal to three. So we can substitute this into our equation to get nine multiplied by three, which we can calculate is equal to 27.
Therefore, if 𝐴 is a matrix of order two by two such that the determinant of 𝐴 is equal to three, then the determinant of three 𝐴 is equal to 27.