Properties of matrices....(1)

Algebra Level 4

Consider a skew symetric matrix A of order m×m (m is a natural number) then if m=2k+1 for some natural number k then which of the following statements are true:-

$\mathbf{1.}$ Trace of A is zero

$\mathbf{2.}$ Determinant of A is zero

$\mathbf{3.}$ Inverse of A does not exist

$\mathbf{4.}$ Transpose of A is equal to additive inverse of A

It depends upon value of k All 4 and 1 only 3 and 2 only

1 solution

Aman Sharma
Oct 22, 2014

Accourding to defination of skew symetric matrix:- A matrix A is called skew symetric if A'=-A hence statement 4 is correct All principle diagnal elements of a skew symetric matrix are zero hence statement 1 is correct Let:- $x=|A|.....(1)$ Also determinant of a matrix is equal to determinent of transpose of matrix:- So $x=|A'|$ $x=|-A|$ $x=(-1)^{m}|A|$ Now accaurding to question m=2k+1 so it shows that m is an odd number hence $x=-|A|....(2)$ Adding (1) and (2):- 2x=0 so x=0 hence determinent of A is zero Hence statement 2 is correct Since dererminent of A is zero hence its inverse does not exist

So all statements are true

Properties of matrices....(1)

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