If \left[\begin{array}{cc}2 & 0\ 5 & 4]\end{array} = P + Q ...

CBSE Class 12 Math 2023 All Sets Solved Paper

Question : 16 of 20

Marks: +1, -0

{[\table 2, 0; 5, 4]} = P + Q

where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to:

Solution:

Let A =

{[\table 2,0; 5,4]}

Any matrix can be written as the sum of a sum of symmetric matrix and a skew-symmetric matrix.
A =

\;{1}/{2}(A + A^T)+\;{1}/{2}(A -A^T)

Here Q =

\;{1}/{2}(A -A^T) = \;{1}/{2} ({[\table 2,0; 5,4]}-{[\table 2,5; 0,4]})

=\;{1}/{2} ({[\table 0,-5; 5,0]})

∴ Q = {[\table 0,-\;{5}/{2}; \;{5}/{2},0]}

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