# Maths-1998-Set II

**Q2) Express the matrix
A = 3 1
-4 -1
as the sum of a symmetric and a skew symmetric matrix.**

Ans2) We know for a square matrix A. 1/2(A + A’ ) is sym and 1/2(A – A’ ) is skew-symmetric. Also

A = 1/2 (A + A’ ) + 1/2 (A – A’ )

Now

A =

3 1

-4 -1

=>A’ =

3 -4

1 -1

… 1/2 (A + A’ ) = 1/2

6 -3

-3 -2

1/2(A – A’ ) = 1/2

0 5

-5 0

Thus A =

0 -3/2

-3/2 -1

+

0 5/2

-5/2 0

where first one is symmetric and the second skew symmetric

**Q3) Evaluate
lim
x->/2 (2x – )/cos x**

Ans3)

lim

x->/2 (2x – )/cos x putting /2 – x = , as x -> /2 , -> 0

= lim

->0 (2(/2 – ) – )/cos (/2 – )

= lim

->0

-2/sin = -2 (as lim /Sin = 1)

->0