Question

For what value of $x$ , is the matrix $A$ = \left[ {\begin{array}{*{20}{c}} 0&1&{ - 2} \\\ { - 1}&0&3 \\\ x&{ - 3}&0 \end{array}} \right] a skew symmetric matrix ?

Answer 1

For a matrix to be skew symmetric the condition which is to be satisfied is $A = - {A^T}$. Here ${A^T}$ denotes the transpose of the matrix $A$ . Here we know the matrix $A$ . From this we can calculate ${A^T}$ and $- {A^T}$. Then putting the value of $A$ and $- {A^T}$ in the above condition we can obtain the value of $x$ .
FORMULA USED :
For a skew symmetric matrix $A = - {A^T}$.

Complete answer: We are given that $A$ = \left[ {\begin{array}{*{20}{c}} 0&1&{ - 2} \\\ { - 1}&0&3 \\\ x&{ - 3}&0 \end{array}} \right] we have to find the value of $x$ such that the matrix satisfies the condition for the skew symmetric matrix .
We know the condition for the skew symmetric matrix $A = - {A^T}$. First we will obtain the value of ${A^T}$ .
Transpose of a matrix is obtained by interchanging rows and columns of matrix, in other words we find the transpose of a matrix by changing rows to columns and columns to rows .
Every element of the matrix $A$ can be denoted by ${a_{ij}}$ where $i$ denotes row number $j$ denotes column number . ${a_{ij}}$ = element of ${i^{th}}$row and ${j^{th}}$ column .
Every element of the matrix ${A^T}$ can be denoted by ${a_{ji}}$ .
$A$ = \left[ {\begin{array}{*{20}{c}} 0&1&{ - 2} \\\ { - 1}&0&3 \\\ x&{ - 3}&0 \end{array}} \right]
By changing rows into column and vice versa we get
${A^T}$= \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&x; \\\ 1&0&{ - 3} \\\ { - 2}&3&0 \end{array}} \right]
So $- {A^T}$= \left[ {\begin{array}{*{20}{c}} 0&1&{ - x} \\\ { - 1}&0&3 \\\ 2&{ - 3}&0 \end{array}} \right]
Putting the values of $A$ and $- {A^T}$ in condition we get

0&1&{ - 2} \\\ { - 1}&0&3 \\\ x&{ - 3}&0 \end{array}} \right]$$=$$\left[ {\begin{array}{*{20}{c}} 0&1&{ - x} \\\ { - 1}&0&3 \\\ 2&{ - 3}&0 \end{array}} \right]$$ Now we have to equate the matrix. While equating the matrices we must equate every element in a row of a column with the corresponding value of the other matrix. $${a_{13}}$$ = element of $${1^{st}}$$row and $${3^{rd}}$$ column . $${a_{31}}$$ = element of $${3^{rd}}$$row and $${1^{st}}$$column . Equating $${a_{13}}$$ we get $$ - x = - 2$$ So $$x = 2$$ Equating $${a_{31}}$$ we get $$x = 2$$ This is the desired answer . **Therefore for $$x = 2$$ matrix $$A$$ is a skew symmetric matrix .** **Note:** Students should remember while multiplying any constant to a matrix we have multiplied this constant to every element of the matrix . Transpose of the matrix should be obtained carefully . Negative sign in the condition is very important ; without the negative sign the condition will become the condition for the symmetric matrix .