Let `|[x^2+3x,x-1,x+3],[x+1,-2x,x-4],[x-3,x+4, 3x]|=a x^4+b x^3+c x^2+e` be an identity in `x ,` where `a , b , c , d , e` are independent of `xdot` Then the value of `e` is (a) 4 (b) 0 (c) 1 (d) none of these

Options

A. 4
B. 0
C. 1
D. none of these

Answer

B

Explanation

Clearly, e is the value of L.H.S. of the given identity at x = 0
For x = 0, We obtain
`LHS = |(0,-1,3),(1,0,-4),(-3,4,0)|`
Note 1 Only square matrices have determinants. The matrices which are not square do not have determinants
Note2 The determinant of a square matrix of order 3 can be expanded along any row or column.
Note 3 If a row or a column of a determinant consists of all zeros, then the value of the determinant is zero.
To evaluate the determinant of a square matrix of order 4 or more we follow the same procedure as discussed in evaluating the determinant of a square matrix of order 3.