Question

express the following matrix as the sum of symmetric and skew symmetric
|2 3 - 1 |
|5 - 2 1 |
|-4 - 5 3|​

Answer 1

Answer:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

(i) [

3

1

5

−1

]

(ii)

⎣

⎢

⎢

⎡

6

−2

2

−2

3

−1

2

−1

3

⎦

⎥

⎥

⎤

(iii)

⎣

⎢

⎢

⎡

3

−2

−4

3

−2

−5

−1

1

2

⎦

⎥

⎥

⎤

(iv) [

1

−1

5

2

]

Medium

Answer 2

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \begin{gathered}\sf \left[\begin{array}{ccc}2&3& - 1\\5& - 2&1\\ - 4&-5&3\end{array}\right]\end{gathered}$

We know,

$\rm :\longmapsto\:A = \dfrac{1}{2}(2A)$

$\rm \: = \: \:\dfrac{1}{2}(A + A)$

$\rm \: = \: \:\dfrac{1}{2}(A + A + {A}^{T} - {A}^{T} )$

$\rm \: = \: \:\dfrac{1}{2}\bigg((A + {A}^{T}) + (A - {A}^{T})\bigg)$

$\rm \: = \: \:\dfrac{1}{2}(A + {A}^{T}) + \dfrac{1}{2} (A - {A}^{T})$

$\rm \: = \: \:P + Q$

$\rm :\longmapsto\:where \: P = \dfrac{1}{2}(A + {A}^{T})$

and

$\rm :\longmapsto\:where \: Q = \dfrac{1}{2}(A - {A}^{T})$

We have

$\rm :\longmapsto\: \begin{gathered}\sf A=\left[\begin{array}{ccc}2&3& - 1\\5& - 2&1\\ - 4&-5&3\end{array}\right]\end{gathered}$

So,

$\rm :\longmapsto\: {A}^{T} = \begin{gathered}\sf \left[\begin{array}{ccc}2&5& - 4\\3& - 2& - 5\\ - 1&1&3\end{array}\right]\end{gathered}$

Now,

Consider,

$\rm :\longmapsto\:A + {A}^{T}$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}2&5& - 4\\3& - 2& - 5\\ - 1&1&3\end{array}\right]\end{gathered} + \begin{gathered}\sf \left[\begin{array}{ccc}2&3& - 1\\5& - 2&1\\ - 4&-5&3\end{array}\right]\end{gathered}$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}4&8& - 5\\8& - 4& - 4\\ - 5&-4&6\end{array}\right]\end{gathered}$

So,

$\rm :\longmapsto \: P = \dfrac{1}{2}(A + {A}^{T})$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}2&4& - \dfrac{5}{2} \\ \\4& - 2& - 2 \\ \\ - \dfrac{5}{2} & - 2& - 3\end{array}\right]\end{gathered}$

Now,

$\rm :\longmapsto\: {P}^{T} = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}2&4& - \dfrac{5}{2} \\ \\4& - 2& - 2 \\ \\ - \dfrac{5}{2} & - 2& - 3\end{array}\right]\end{gathered} = P$

$\rm :\implies\:P \: is \: symmetric.$

Now,

Consider

$\rm :\longmapsto\:A - {A}^{T}$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}2&3& - 1\\5& - 2&1\\ - 4&-5&3\end{array}\right]\end{gathered} - \begin{gathered}\sf \left[\begin{array}{ccc}2&5& - 4\\3& - 2& - 5\\ - 1&1&3\end{array}\right]\end{gathered}$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}0& - 2& 3\\ 2& 0& 6\\ - 3& - 6&0\end{array}\right]\end{gathered}$

So,

$\rm :\longmapsto \: Q = \dfrac{1}{2}(A - {A}^{T})$

$\rm \: = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}0& - 1& \dfrac{3}{2} \\ \\ 1& 0& 3 \\ \\ - \dfrac{3}{2} & - 3& 0\end{array}\right]\end{gathered}$

Now,

$\rm :\longmapsto\: {Q}^{T} = \: \:\begin{gathered}\sf \left[\begin{array}{ccc}0& 1& - \dfrac{3}{2} \\ \\ - 1& 0& - 3 \\ \\ \dfrac{3}{2} & 3& 0\end{array}\right]\end{gathered}$

$\rm \: = \: \: - \begin{gathered}\sf \left[\begin{array}{ccc}0& - 1& \dfrac{3}{2} \\ \\ 1& 0& 3 \\ \\ - \dfrac{3}{2} & - 3& 0\end{array}\right]\end{gathered}$

$\rm \: = \: \: - \: Q$

$\bf\implies \:Q \: is \: skew \: symmetric.$

Thus, we have

$\rm :\longmapsto\:A = P + Q$

$\rm :\longmapsto\:\begin{gathered}\sf \left[\begin{array}{ccc}2&3& - 1\\5& - 2&1\\ - 4&-5&3\end{array}\right]\end{gathered}$

$\rm \: = \begin{gathered}\sf \left[\begin{array}{ccc}2&4& - \dfrac{5}{2} \\ \\4& - 2& - 2 \\ \\ - \dfrac{5}{2} & - 2& - 3\end{array}\right]\end{gathered} + \begin{gathered}\sf \left[\begin{array}{ccc}0& - 1& \dfrac{3}{2} \\ \\ 1& 0& 3 \\ \\ - \dfrac{3}{2} & - 3& 0\end{array}\right]\end{gathered}$

Additional Information :-

1. If A and B are symmetric matrices then

• AB + BA is symmetric matrix.

• AB - BA is skew - symmetric.

2. The main diagonal elements of skew - symmetric matrix are always zero.

3. The sum of all the elements of skew symmetric matrix are always zero.

4. The determinant of skew symmetric matrix of odd order is 0.