Question

Let S = `{((a_11, a_12),(a_21, a_22)): a_(ij) ∈ {0, 1, 2}, a_11 = a_22}`

Then the number of non-singular matrices in the set S is ______.

Options

• 27

• 24

• 10

• 20

Answer 1

Let S = `{((a_11, a_12),(a_21, a_22)): a_(ij) ∈ {0, 1, 2}, a_11 = a_22}`

Then the number of non-singular matrices in the set S is 20.

Explanation:

The matrices in the form

`[(a_11, a_12),(a_21, a_22)], a_(ij) ∈ {0, 1, 2}, a_11 = a_22` are 

`[(0, 0//1//2),(0//1//2, 0)], [(1, 0//1//2),(0//1//2, 1)], [(2, 0//1//2),(0//1//2, 2)]`

At any place, 0/1/2 means 0, 1 or 2 will be the element at that place.

Hence there are total of 27 (= 3 × 3 + 3 × 3 + 3 × 3) matrices of the above form. Out of which the matrices which are singular are

`[(0, 0//1//2),(0, 0)], [(0, 0),(1//2, 0)], [(1, 1),(1, 1)], [(2, 2),(2, 2)]`

Hence there are total 7(= 3 + 2 + 1 + 1) singular matrices.

Therefore number of all non-singular matrices in the given form = 27 – 7 = 20