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

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