First slide

Applications of determinant

Question

The inverse of a skew-symmetric matrix of odd order is

Moderate

A

a symmetric matrix
B

a skew-symmetric matrix
C

diagonal matrix
D

does not exist

Solution

Let A be a skew-symmetric, matrix of order n. By definition
$\begin{array}{l} A^{'} = - A \\ \Rightarrow |A^{'}| = | - A | \Rightarrow | A | = (- 1)^{n} | A | \end{array}$
$\begin{array}{lrl} \Rightarrow | A | = - | A | \\ \Rightarrow 2 | A | = 0 \Rightarrow | A | = 0 \end{array}$ [∵ n is odd]
$∴ A^{- 1}$ does not exist.

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