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

If det (π΄π΅) = 18 and det (π΄) = 2, find det (π΅).

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### Video Transcript

If the determinant of π΄ times π΅ is equal to 18 and the determinant of π΄ is equal to two, find the determinant of π΅.

In this question, weβre given some information about two matrices, π΄ and π΅. Weβre told the determinant of matrix π΄ multiplied by matrix π΅ is 18 and the determinant of matrix π΄ is equal to two. We need to use this information to determine the determinant of π΅. To answer this question, letβs start by recalling the property of determinants, which links the products of matrices with the determinant of each individual matrix.

We recall if π΄ and π΅ are square matrices of the same order, then the determinant of π΄ times π΅ is equal to the determinant of π΄ multiplied by the determinant of π΅. We might be tempted to apply this property directly to our question. However, thereβs a problem; weβre not told the orders of matrices π΄ and π΅. And to apply this property, we do need the matrices π΄ and π΅ are square matrices of the same order. However, we can show that this must be the case from the information given. First, weβre told the determinant of matrix π΄ is equal to two. And we recall we can only find the determinant of square matrices, so π΄ is a square matrix. Similarly, the determinant of π΄π΅ is equal to 18, so π΄ times π΅ is also a square matrix.

We can use this information to find an expression for the order of matrix π΅. First, since π΄ is a square matrix, letβs start by saying its order is of the form π by π. Next, we donβt know the order of matrix π΅. Letβs say its order is π by π. Then, there are two different ways of finding an expression for the order of π΄ times π΅. 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, π΄π΅ must have order π by π. However, weβve already shown that π΄π΅ is a square matrix. So, the number of rows must be equal to the number of columns. In other words, π is equal to π. So, we can replace π with π.

Finally, to determine the value of π, we notice weβre allowed to multiply π΄ on the right by matrix π΅. 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, π must be equal to π. Therefore, weβve shown π΄ is a square matrix, π΅ is a square matrix, and the orders of matrices π΄ and π΅ are equal. So, we can just apply our property to the question. We have the determinant of π΄π΅ is equal to the determinant of π΄ multiplied by the determinant of π΅. Next, weβre told in the question the determinant of π΄π΅ is 18 and the determinant of π΄ is two. So, 18 is equal to two times the determinant of π΅. Finally, we can divide both sides of the equation through by two to get the determinant of π΅ is nine.

Therefore, we were able to show if the determinant of π΄π΅ is 18 and the determinant of π΄ is two, then the determinant of π΅ must be equal to nine.