Express following matrices as sum of a symmetric and a skew symmetric matrix : left[begi

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

medium

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Option A

Option B

Option C

Option D

Solution

Let $A=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right],$ then $A^{\prime}=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Now, $A+A^{\prime}=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right] $ $+\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right] $ $=\left[\begin{array}{ccc}12 & -4 & 4 \\ -4 & 6 & -2 \\ 4 & -2 & 6\end{array}\right]$

Let $P=\frac{1}{2}\left(A+A^{\prime}\right)=\frac{1}{2}\left[\begin{array}{ccc}12 & -4 & 4 \\ -4 & 6 & -2 \\ 4 & -2 & 6\end{array}\right]=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Now, $P^{\prime}=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]=P$

Thus, $P=\frac{1}{2}\left(A+A^{\prime}\right)$ is a symmetric matrix.

Now, $A-A=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]+\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

Let $Q=\frac{1}{2}\left(A-A^{\prime}\right)=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

Now, $Q^{\prime}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]=-Q$

Thus $Q=\frac{1}{2}\left(A-A^{\prime}\right)$ is a skew-symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$

$P+Q=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$ $+\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$ $=\left[\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]=A$

Standard 12

Mathematics

For any square matrix $A$, $A{A^T}$ is a

normal

If $\mathrm{A}=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right],$ then verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.

easy

A square matrix $A = [{a_{ij}}]$ in which ${a_{ij}} = 0$ for $i\, \ne j$ and ${a_{ij}} = k$ (constant) for $i = j$ is called a

normal

Let three matrices $A =$$\left[ {\begin{array}{{20}{c}}2&1\\4&1\end{array}} \right]$ ; $B =$$\left[ {\begin{array}{{20}{c}}3&4\\2&3\end{array}} \right]$ and $C =$$\left[ {\begin{array}{*{20}{c}}3&{ – 4}\\{ – 2}&3\end{array}} \right]$ then $T_r(A) + t_r$ $\left( {\frac{{ABC}}{2}} \right)$ $+$ $t_r$$\left( {\frac{{A{{(BC)}^2}}}{4}} \right)$ $+$ $t_r$ $\left( {\frac{{A{{(BC)}^3}}}{8}} \right)$ $+$ ……. $+ \infty =$

normal

Show that the matrix $B ^{\prime}A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

medium

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Solution

Similar Questions

For any square matrix $A$, $A{A^T}$ is a

If $\mathrm{A}=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right],$ then verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.

A square matrix $A = [{a_{ij}}]$ in which ${a_{ij}} = 0$ for $i\, \ne j$ and ${a_{ij}} = k$ (constant) for $i = j$ is called a

Show that the matrix $B ^{\prime}A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

Start a Free Trial Now