Let A be the `2 xx 2` matrix given by `A=[a_("ij")]` where `a_("ij") in {0, 1, 2, 3, 4}` such theta `a_(11)+a_(12)+a_(21)+a_(22)=4` then which of the following statement(s) is/are true ?

Options

A. Number of matrices A such that the trace of A equal to 4, is 5
B. Number of matrices A, such that A is invertible is 18
C. Absolute difference between maximum value and minimum value of det (A) is 8
D. Number of matrices A such that A is either symmetric (or) skew symmetric and det (A) is divisible by 2, is 5.

Explanation

(1) Possible matrices are
`[(1,0),(0,3)], [(3,0),(0,1)], [(2,0),(0,2)], [(4,0),(0,0)], [(0,0),(0,4)]`
(2) Using 0, 0, 2, 2, there are two matrices which are invertible which are `[(2,0),(0,2)]` and `[(0,2),(2,0)]`
Using 0, 0, 1, 3, there are four matrices which are invertible.
Using 0, 1, 1, 2, there are twelve matrices which are invertible.
Using 0, 0, 0, 4 and using 1, 1, 1, 1 no matrix is formed, which isinvertiable.
`:.` Total number of matrices `= 18`
(3) `|4-(-4)|=8`
(4) There are five matrices, which are either symmetric or skew symetric and whose determinant is divisible by 2. These are `[(2,0),(0,2)], [(0,2),(2,0)], [(0,0),(0,4)], [(4,0),(0,0)], [(1,1),(1,1)]`