UPSC NDA Math Model Paper 8

Given: A and B are square matrices of order 3 such that |A| = -1, |B| = 3
Here, we have to find the value of |3 AB|
As we know that, if A and B are two determinants of order n, then |A ⋅ B| = |A| ⋅ |B|
⇒ |3 AB| = |3A| ⋅ |B|
As we know that, if A is a matrix of order n, then |k.A|=kn.|A|, where k∈R
Here n = 3 So, |3A| = 33 ⋅ |A|
⇒ |3 AB| = 33 ⋅ |A| ⋅ |B|
Now by substituting |A| = -1, |B| = 3 in the above equation we get,
⇒ |3 AB| = 27 ⋅ (- 1) ⋅ (3) = - 81
Hence, option B is the correct answer.