Question

Sum of two skew symmetric matrices is always ________ matrix.

Answer 1

A skew symmetric matrix is a square matrix whose transpose is its negation, i.e., it satisfies the condition ${M^T} = - M$, where ${M^T}$ is the transpose of a matrix $M$. We will consider two skew symmetric matrices $A$ and $B$ and we will take transpose of $\left( {A + B} \right)$ i.e., $\left( {A + B} \right)'$ if the result comes out to be $- \left( {A + B} \right)$ then it is a skew symmetric matrix.

Complete step by step answer:
To find what is the sum of two skew symmetric matrices, we need to first understand the transpose of a matrix and a skew symmetric matrix. Let $M$ be a matrix of order $m \times n$, then the $n \times m$ matrix obtained by interchanging the rows and the columns of $M$ is called the transpose of $M$ and is denoted by ${M^T}$. A square matrix is said to be skew symmetric if the transpose of the matrix equals its negative. A matrix $M$ of order $m \times n$ is said to be skew symmetric if and only if ${a_{ij}} = - {a_{ji}}$ where $i = row{\text{ }}entry$ and $j = column{\text{ }}entry$.

For any skew symmetric matrix $M$, ${M^T} = - M$. Let two skew symmetric matrices, $A$ and $B$.
$A' = - A$ and $B' = - B - - - (1)$
Now, we will take the transpose of the sum of $A$ and $B$ i.e., $\left( {A + B} \right)'$.
As we know from the properties of the transpose of a matrix that $\left( {A + B} \right)' = A' + B'$. Therefore, we get
$\Rightarrow \left( {A + B} \right)' = A' + B'$
Using $(1)$, we get
$\Rightarrow \left( {A + B} \right)' = - A - B$
$\therefore \left( {A + B} \right)' = - \left( {A + B} \right)$

Therefore, the sum of two skew symmetric matrices is always a skew symmetric matrix.

Note: A symmetric and a skew symmetric matrix, both are square matrices. The diagonal element of a skew symmetric matrix is equal to zero and therefore the sum of elements in the main diagonal of a skew symmetric matrix is equal to zero. Also note that the determinant of the skew symmetric matrix is non-negative.