Express following matrices as sum of a symmetric and a skew symmetric matrix : left[begin{arra

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3 and 4 .Determinants and Matrices

medium

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right]$

Option A

Option B

Option C

Option D

Solution

Let $A=\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right],$ then $A^{\prime}=\left[\begin{array}{cc}3 & 1 \\ 5 & -1\end{array}\right]$

Now, $A+A^{\prime}=\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right]+\left[\begin{array}{cc}3 & 1 \\ 5 & -1\end{array}\right]=\left[\begin{array}{cc}6 & 6 \\ 6 & -2\end{array}\right]$

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

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

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

Now, $A-A^{\prime}=\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right]-\left[\begin{array}{cc}3 & 1 \\ 5 & -1\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -4 & 0\end{array}\right]$

Let $Q=\frac{1}{2}\left(A-A^{\prime}\right)=\frac{1}{2}\left[\begin{array}{cc}0 & 4 \\ -4 & 0\end{array}\right]=\left[\begin{array}{cc}0 & 2 \\ -2 & 0\end{array}\right]$

Now, $Q^{\prime}=\left[\begin{array}{cc}0 & 2 \\ -2 & 0\end{array}\right]=-Q$

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

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

$P+Q=\left[\begin{array}{cc}3 & 3 \\ 3 & -1\end{array}\right]$ $+\left[\begin{array}{cc}0 & 2 \\ -2 & 0\end{array}\right]$ $=\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right]$ $=A$

Standard 12

Mathematics

Find the transpose of the following matrices : $\left[\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right]$

easy

If $\mathrm{A}=\left[\begin{array}{c}-2 \\ 4 \\ 5\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}1 & 3 & -6\end{array}\right],$ verify that $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$.

medium

Let $A=\left\{X=(x, y, z)^{T}: P X=0\right.$ and $\left.\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\right\}$ where $\mathrm{P}=\left[\begin{array}{ccc}1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1\end{array}\right]$ then the set $\mathrm{A}$

hard

(JEE MAIN-2020)

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

The matrix $A = \left[ {\begin{array}{*{20}{c}}1&{ – 3}&{ – 4}\\{ – 1}&{\,\,\,3}&{\,\,4}\\1&{ – 3}&{ – 4}\end{array}} \right]$ is nilpotent of index