Question

Express the matrix A as the sum of a symmetric and skew symmetric matrix where A=[4 5 6,/-1 0 1,/ 2 1 2]​

Answer 1

       $\underline{\bold{Solution:}}$

       $\text{The given matrix is }A = \left[\begin{array}{ccc}4&5&6\\-1&0&1\\2&1&2\end{array}\right]$

$\implies A^{T} = \left[\begin{array}{ccc}4&-1&2\\5&0&1\\6&1&2\end{array}\right]$

       $\text{Adding }A \text{ and }A^{T}$

$\implies A + A^{T} = \left[\begin{array}{ccc}4&5&6\\-1&0&1\\2&1&2\end{array}\right]+\left[\begin{array}{ccc}4&-1&2\\5&0&1\\6&1&2\end{array}\right]$

$\implies A + A^{T} = \left[\begin{array}{ccc}8&4&8\\4&0&2\\8&2&4\end{array}\right]$

       $\text{Multiplying }\dfrac12 \text{ on both sides}$

$\implies \dfrac12(A + A^{T}) = \dfrac12\left[\begin{array}{ccc}8&4&8\\4&0&2\\8&2&4\end{array}\right]$

$\implies \dfrac12(A + A^{T}) = \left[\begin{array}{ccc}4&2&4\\2&0&1\\4&1&2\end{array}\right]$

       $\text{Let }P = \dfrac12(A + A^{T})$

$\implies P = \left[\begin{array}{ccc}4&2&4\\2&0&1\\4&1&2\end{array}\right] \text{ ------ (i)}$

$\implies P^{T} = \left[\begin{array}{ccc}4&2&4\\2&0&1\\4&1&2\end{array}\right]$

       $\text{We can see that }P = P^{T}$

 $\therefore\text{ }\text{ }P \text{ is a symmetric matrix}$

       $\text{Substracting }A^{T} \text{ from }A$

$\implies A - A^{T} = \left[\begin{array}{ccc}4&5&6\\-1&0&1\\2&1&2\end{array}\right]-\left[\begin{array}{ccc}4&-1&2\\5&0&1\\6&1&2\end{array}\right]$

$\implies A - A^{T} = \left[\begin{array}{ccc}0&6&4\\-6&0&0\\-4&0&0\end{array}\right]$

       $\text{Multiplying }\dfrac12 \text{ on both sides}$

$\implies \dfrac12(A - A^{T}) = \dfrac12\left[\begin{array}{ccc}0&6&4\\-6&0&0\\-4&0&0\end{array}\right]$

$\implies \dfrac12(A - A^{T}) = \left[\begin{array}{ccc}0&3&2\\-3&0&0\\-2&0&0\end{array}\right]$

       $\text{Let }Q = \dfrac12(A - A^{T})$

$\implies Q =\left[\begin{array}{ccc}0&3&2\\-3&0&0\\-2&0&0\end{array}\right] \text{ ------ (ii)}$

$\implies Q^{T} = \left[\begin{array}{ccc}0&-3&-2\\3&0&0\\2&0&0\end{array}\right]$

       $\text{We can see that }Q^{T} = -Q$

 $\therefore \text{ }\text{ }Q \text{ is a skew-symmetric matrix}$      

       $\text{Adding }P \text{ and }Q$

$\implies P + Q$

       $\text{We know that }P = \dfrac12(A+A^{T}) \text{ and }Q = \dfrac12(A-A^{T})$

$\implies P + Q = \dfrac12(A+A^{T}) + \dfrac12(A-A^{T})$

$\implies P + Q = \dfrac12(A+A^{T}+ A - A^{T})$

$\implies P + Q = \dfrac12(2A)$

$\implies P + Q = A$

       $\text{From (i) and (ii)}$

$\implies \boxed{\left[\begin{array}{ccc}4&2&4\\2&0&1\\4&1&2\end{array}\right] + \left[\begin{array}{ccc}0&3&2\\-3&0&0\\-2&0&0\end{array}\right] = A}$