Question

If A is a 3 x 3 skew-symmetric matrix, then trace of A is equal to
(a) – 1
(b) 7
(c) \[\left| A \right|\]
(d) None of these

Answer 1

Hint: First of all, assume a 3 x 3 matrix A that is \[\left| \begin{matrix}
   p & q & r \\
   s & t & u \\
   v & w & x \\
\end{matrix} \right|\]. Now, write its transpose \[{{A}^{T}}\] by interchanging its rows and columns. Now use \[{{A}^{T}}=-A\] as given matrix is a skew-symmetric matrix and from this, find the diagonal denoted by equating them and find their sum which is the trace of the matrix.

Complete step-by-step answer:
In this question, we are given that A is a 3 x 3 matrix, then we have to find the trace of A. Before proceeding with the question, let us understand a few terms.
Transpose of a Matrix: Transpose of a matrix is a matrix formed by interchanging the rows and columns of the given matrix. If we are given a 2 x 3 matrix as \[A=\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right|\], then its transpose would be \[{{A}^{T}}=\left| \begin{matrix}
   a & d & g \\
   b & e & h \\
   c & f & i \\
\end{matrix} \right|\].
Skew Symmetric Matrix: If the transpose of a matrix is equal to the negative of the given matrix itself, then the matrix is said to be a skew symmetric matrix. This means that for a matrix to be skew-symmetric, \[{{A}^{T}}=-A\].
Trace of a Matrix: Trace of a matrix is equal to the sum of the elements on the main diagonal of the given matrix. If we are given a matrix \[A=\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right|\], then its trace would be T = a + e + i.
Now, let us consider our question. Let us assume a 3 x 3 matrix A which is as follows.
\[A=\left| \begin{matrix}
   p & q & r \\
   s & t & u \\
   v & w & x \\
\end{matrix} \right|....\left( i \right)\]
By interchanging the row and column of the above matrix, we get the transpose of the above matrix as,
\[{{A}^{T}}=\left| \begin{matrix}
   p & s & v \\
   q & t & w \\
   r & u & x \\
\end{matrix} \right|....\left( ii \right)\]
We know that for a matrix A to be skew-symmetric,
\[{{A}^{T}}=-A\]
By substituting the value of A and \[{{A}^{T}}\] from equation (i) and (ii), we get,
\[\left| \begin{matrix}
   p & s & v \\
   q & t & w \\
   r & u & x \\
\end{matrix} \right|=-\left| \begin{matrix}
   p & q & r \\
   s & t & u \\
   v & w & x \\
\end{matrix} \right|\]
By taking the negative sign inside the matrix, we get,
\[\left| \begin{matrix}
   p & s & v \\
   q & t & w \\
   r & u & x \\
\end{matrix} \right|=\left| \begin{matrix}
   -p & -q & -r \\
   -s & -t & -u \\
   -v & -w & -x \\
\end{matrix} \right|\]
We know that when two matrices are equal, all their corresponding elements are also equal. So, we get,
p = – p
p + p = 0
2p = 0
p = 0…..(iii)
Also, t = – t
t + t = 0
2t = 0
t = 0….(iv)
Also, x = – x
x + x = 0
2x = 0
x = 0…..(v)
Now, we know that the trace of a matrix is equal to the sum of the elements of its main diagonal. So, we get,
Trace of matrix A = p + t + x
By substituting the values of p, t and x from equation (iii), (iv) and (v), we get
Trace of matrix A = 0 + 0 + 0 = 0
Hence, option (d) is the right answer.
Note: Students must note that any skew-symmetric matrix is always in the form \[\left| \begin{matrix}
   0 & a & b \\
   -a & 0 & c \\
   -b & -c & 0 \\
\end{matrix} \right|\] where a, b, and c can take any values. Also, note that unlike in determinants, when we multiply a matrix with a number, that number is multiplied by all of its elements and not just by a single row or column. So, this must be taken care of. Always remember that for a matrix to be symmetric or skew-symmetric, it must be a square matrix.