Question

Sum of two skew symmetric matrices is always ________ matrix.

Answer 1

Hint: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.