Matrices

Consider the matrix f(x)= bmatrix x & - x & 0\ x & x & 0\ 0 & 0 & 1 bmatrix. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

Answer

(3) Both Statement I and Statement II are true

Solution

Multiply (use (x+y) and (x+y) formulae):

f(x)f(y)= bmatrix (x+y) & - (x+y) & 0\ (x+y) & (x+y) & 0\ 0 & 0 & 1 bmatrix=f(x+y)

Statement II is true

Put y=-x: f(x)f(-x)=f(0)=I f(-x)= [f(x) ]-1 Statement I is true

Answer: (3)