Question

If A and B are square matrices of order 3 such that $$\left|A\right|=-1$$ , $$\left|B\right|=3$$ , then the determinant of 3AB is
A. -9
B. -27
C. -81
D. 81

Answer 1

Hint: For any square matrix of order n, we know that if we multiply it with any constant K then $$\left|KA\right|=K^n\left|A\right|$$ . So we will use this property of matrix and determinants to solve this question.

Complete step by step answer:
For this Question we will use the property of determinants that is $$\left|KA\right|=K^n\left|A\right|$$ where n is the order of matrix
So we are given that $$\left|A\right|=-1,\left|B\right|=3$$ and we are told to find the value of $$\left|3AB\right|$$
We can break $$\left|3AB\right|$$ as $$\left|3A\right|\times \left|B\right|$$
Now we know that both A and B are square matrices of order 3 which means if i want to take 3 out from $$\left|3A\right|$$ It will come out as $$3^3\left|A\right|$$
Now we are left with $$\left|3A\right|\times \left|B\right|=3^3\times \left|A\right|\times \left|B\right|$$
Now we know that $$\left|A\right|=-1\mathrm{\&}\left|B\right|=3$$
So putting the values of $$\left|A\right|\mathrm{\&}\left|B\right|$$ we will get
$$\begin{array}{l}\Rightarrow \left|3AB\right|=27\times (-1)\times 3\\ \Rightarrow \left|3AB\right|=27\times (-3)\\ \Rightarrow \left|3AB\right|=-81\end{array}$$

So, the correct answer is “Option C”.

Note: The thing is to remember that $$\left|KA\right|=K^n\left|A\right|$$ , a lot of students usually forget this relation and can’t solve the question. Also remember that $$\left|AB\right|=\left|A\right|\times \left|B\right|$$ .