Question Video: Finding the Determinant of a Product of Matrices Mathematics

If det (𝐴𝐡) = 18 and det (𝐴) = 2, find det (𝐡).

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Video Transcript

If the determinant of 𝐴 times 𝐡 is equal to 18 and the determinant of 𝐴 is equal to two, find the determinant of 𝐡.

In this question, we’re given some information about two matrices, 𝐴 and 𝐡. We’re told the determinant of matrix 𝐴 multiplied by matrix 𝐡 is 18 and the determinant of matrix 𝐴 is equal to two. We need to use this information to determine the determinant of 𝐡. 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 𝐡 are square matrices of the same order, then the determinant of 𝐴 times 𝐡 is equal to the determinant of 𝐴 multiplied by the determinant of 𝐡. 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 𝐡. And to apply this property, we do need the matrices 𝐴 and 𝐡 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 𝐴𝐡 is equal to 18, so 𝐴 times 𝐡 is also a square matrix.

We can use this information to find an expression for the order of matrix 𝐡. 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 𝐡. Let’s say its order is π‘š by 𝑙. Then, there are two different ways of finding an expression for the order of 𝐴 times 𝐡. 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, 𝐴𝐡 must have order 𝑛 by 𝑙. However, we’ve already shown that 𝐴𝐡 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 π‘š, we notice we’re allowed to multiply 𝐴 on the right by matrix 𝐡. 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 π‘š. Therefore, we’ve shown 𝐴 is a square matrix, 𝐡 is a square matrix, and the orders of matrices 𝐴 and 𝐡 are equal. So, we can just apply our property to the question. We have the determinant of 𝐴𝐡 is equal to the determinant of 𝐴 multiplied by the determinant of 𝐡. Next, we’re told in the question the determinant of 𝐴𝐡 is 18 and the determinant of 𝐴 is two. So, 18 is equal to two times the determinant of 𝐡. Finally, we can divide both sides of the equation through by two to get the determinant of 𝐡 is nine.

Therefore, we were able to show if the determinant of 𝐴𝐡 is 18 and the determinant of 𝐴 is two, then the determinant of 𝐡 must be equal to nine.

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