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

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 : $\left[\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right]$

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

Express the matrix $\mathrm{B}=\left[\begin{array}{rrr}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$ as the sum of a symmetric and a skew symmetric matrix.

medium

Find $\frac{1}{2}\left(A+A^{\prime}\right)$ and $\frac{1}{2}\left(A-A^{\prime}\right),$ when $A=\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right]$

hard

The matrix $A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ – 1/\sqrt 2 }&{ – 1/\sqrt 2 }\end{array}} \right]$ is

easy

Match the Statements / Expressions in Column $I$ with the Statements / Expressions in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.

Column $I$ Column $II$

$(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is $(p)$ $0$

$(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are $(q)$ $1$

$(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than $(r)$ $2$

$(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are $(s)$ $3$

hard

(IIT-2008)

If $A$ is a skew symmetric matrix and $n $ is a positive integer, then ${A^n}$ is

normal

Column $I$	Column $II$
$(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is	$(p)$ $0$
$(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are	$(q)$ $1$
$(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than	$(r)$ $2$
$(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are	$(s)$ $3$