mathematics-class-12-unit-03-chapter-07-matrix-and-determinant-l-1-5-m-icgee64ti

### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
### <primary style="font-size:24px"> Matrix transpose </primary>
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- Let $A$ be an $m \times n$ matrix defined by

- $A=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\\ a_{21} & a_{22} & \cdots & a_{2 n} \\\ \vdots & & &  \\\a_{m 1} & a_{m 2} & \cdots & a_{m n}\end{array}\right]$

- A=$\left[a_{i j}\right]$

- Transpose of $A$ is denoted by $A^{\top}$
- and it in defined by

- $A^{\top}=\left[\begin{array}{ccc}a_{11} & a_{21} & -a_{m 1} \\\a_{12} & a_{22} & a_{m 2} \\\ \vdots & \vdots & \vdots \\\a_{1 n} & a_{2 n} & a_{m n}
\end{array}\right]$

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<footer style = "font-size: 18px"> $\rightarrow$ Matrix-and-determinant lecture-7 $\rightarrow$ <span style = "color:blue"> Matrix transpose </span> $\rightarrow$ Symmetric and skew-symmetric matrices $\rightarrow$ Symmetric and skew-symmetric matrices </footer>

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<p>when $m=n$, we call $A$ as square matrix.</p>
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<p>let $A$ be an $n \times n$ square matrix</p>
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<li>$A$ is said to be symmetric of</li>
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<li>$ {A}^{\top}=A \text {, ie., } a_{i j}=a_{j i} \forall i, j $</li>
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<li>$A$ is said to be skew-symmetric</li>
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<footer style = "font-size: 18px">Matrix-and-determinant lecture-7 $\rightarrow$ Matrix transpose $\rightarrow$ <span style = "color:blue"> Symmetric and skew-symmetric matrices </span> $\rightarrow$ Symmetric and skew-symmetric matrices $\rightarrow$ Properties of matrix transpose </footer>

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<li>Let $A$ be an $n \times n$ matrix, determinant $A$ is a scalar quantity. We also denote determinant of $A$ by $|A|$</li>
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<footer style = "font-size: 18px">Symmetric and skew-symmetric matrices $\rightarrow$ Properties of matrix transpose $\rightarrow$ <span style = "color:blue"> Determinant of matrix </span> $\rightarrow$ Determinant of 2×2 and 3×3 $\rightarrow$ Determinant of 3×3 </footer>

### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
### <primary style="font-size:24px"> Determinant of 2×2 and 3×3 </primary>
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- $ n=2, \quad A  =\left[\begin{array}{ll}a_{11} & a_{12} \\\ a_{21} & a_{22}\end{array}\right] $

- $|A|  =a_{11} \cdot a_{22}-a_{21} a_{12}$

- $n=3, \quad A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{array}\right]$

- $\begin{aligned} & |A|=a_{11}\left|\begin{array}{ll}a_{22} & a_{23} \\\ a_{32} & a_{33}\end{array}\right|-a_{12}\left|\begin{array}{ll}a_{21} & a_{23} \\\ a_{31} & a_{33}\end{array}\right| +a_{13}\left|\begin{array}{ll}a_{21} & a_{22} \\\ a_{31} & a_{32}\end{array}\right| \\\ & \end{aligned}$

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<footer style = "font-size: 18px">Properties of matrix transpose $\rightarrow$ Determinant of matrix $\rightarrow$ <span style = "color:blue"> Determinant of 2×2 and 3×3 </span> $\rightarrow$ Determinant of 3×3 $\rightarrow$ Properties of determinant </footer>

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- sign of $a_{i j}$ is given by the sign of $(-i)^{i+j}$

- $ A=\left[\begin{array}{lll} a_{12} & a_{12} & a_{13} \\\a_{21} & a_{22} & a_{23} \\\a_{31} & a_{32} & a_{33}
\end{array}\right] $

- $|A|=a_{31}\left|\begin{array}{ll}a_{12} & a_{13} \\\ a_{22} & a_{23}\end{array}\right|-a_{32}\left|\begin{array}{ll}a_{11} & a_{13} \\\ a_{21} & a_{23}\end{array}\right| +a_{33}\left|\begin{array}{ll}a_{11} & a_{12} \\\ a_{21} & a_{22}\end{array}\right|$

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<footer style = "font-size: 18px">Determinant of matrix $\rightarrow$ Determinant of 2×2 and 3×3 $\rightarrow$ <span style = "color:blue"> Determinant of 3×3 </span> $\rightarrow$ Properties of determinant $\rightarrow$ Properties of determinant cont... </footer>

### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
### <primary style="font-size:24px"> Properties of determinant </primary>
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- i) $\operatorname{det} A=\operatorname{det} A^{\top}$

- ii) If any two rows (or columns) of a determinant are inter changed them sign of determinant changes.

- iii) If all the elements of a row (or column) of $A$ ore zero,
- $ \operatorname{det} A=0 $

- iv) If any two rows (or columns) of a matrix $A$ are identical, $\operatorname{det} A=0$

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<footer style = "font-size: 18px">Determinant of 2×2 and 3×3 $\rightarrow$ Determinant of 3×3 $\rightarrow$ <span style = "color:blue"> Properties of determinant </span> $\rightarrow$ Properties of determinant cont... $\rightarrow$ Properties of determinant cont... </footer>

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<p>v)</p>
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<li>
<p>$\quad\left|\begin{array}{ccc}k a_{11} & k a_{12} & k a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=k\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$</p>
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<p>vi)</p>
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<p>$\begin{aligned}\left|\begin{array}{ccc}a_{11} & a_{12}+x & a_{13} \\ a_{21} & a_{22}+y & a_{23} \\ a_{31} & a_{32}+z & a_{33}\end{array}\right| & =\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right| +\left|\begin{array}{lll}a_{12} & x & a_{13} \\ a_{21} & y & a_{23} \\ a_{31} & z & a_{33}\end{array}\right|\end{aligned}$</p>
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<footer style = "font-size: 18px">Determinant of 3×3 $\rightarrow$ Properties of determinant $\rightarrow$ <span style = "color:blue"> Properties of determinant cont... </span> $\rightarrow$ Properties of determinant cont... $\rightarrow$ Inverse of matrix </footer>

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<p>Let $A$ be an $n \times n$ matrix its inverse is defined by $A^{-1}=\frac{\operatorname{adj} A}{\operatorname{det} A}$ where $\operatorname{det} A \neq 0$ $\Rightarrow A^{-1}$ exists only if $\operatorname{det} A \neq 0$</p>
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<p>We call an invertible matrix as non-singular matrix.</p>
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<footer style = "font-size: 18px">Properties of determinant cont... $\rightarrow$ Properties of determinant cont... $\rightarrow$ <span style = "color:blue"> Inverse of matrix </span> $\rightarrow$ Adjoint of a matrix $\rightarrow$ Adjoint of a matrix contd... </footer>

### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
### <primary style="font-size:24px"> Adjoint of a matrix </primary>
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- $\operatorname{adj} A=$ transpose of co-factor matrix.

- $\begin{aligned} A & =\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{array}\right] \\\ \operatorname{adj} A & =\left[\begin{array}{lll}A_{11} & A_{21} & A_{31} \\\ A_{12} & A_{22} & A_{32} \\\ A_{13} & A_{23} & A_{33}\end{array}\right]\end{aligned}$

- $\operatorname{adj} A=\left[A_{i j}\right]^{\top}$

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<footer style = "font-size: 18px">Properties of determinant cont... $\rightarrow$ Inverse of matrix $\rightarrow$ <span style = "color:blue"> Adjoint of a matrix </span> $\rightarrow$ Adjoint of a matrix contd... $\rightarrow$ Determinant of adjoint of a matrix </footer>

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### <primary style="font-size:24px"> Adjoint of a matrix contd... </primary>
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- $A_{i j}=(-1)^{i+j} \times$ determinant of sub matrix obtained by deleting $i^{th}$-row \& j $^{\text {th }}$ column

- e.g.,
- $ A_{11}=\left|\begin{array}{ll} a_{22} & a_{23} \\\a_{32} & a_{33}\end{array}\right| $

- $ A_{12}=-\left|\begin{array}{ll}a_{21} & a_{23} \\\ a_{31} & a_{33}\end{array}\right|$

- Similarly other $A_{i j}$ can be calculated.

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<footer style = "font-size: 18px">Inverse of matrix $\rightarrow$ Adjoint of a matrix $\rightarrow$ <span style = "color:blue"> Adjoint of a matrix contd... </span> $\rightarrow$ Determinant of adjoint of a matrix $\rightarrow$ Problem-1 </footer>

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<p>$ \operatorname{det} A^{-1}=\frac{1}{\operatorname{det} A} $</p>
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<p>$A A^{-1}=I$</p>
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<p>$\operatorname{det}\left(A A^{-1}\right)\operatorname{det} I =1 $</p>
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<p>$ \operatorname{det}  A \cdot \operatorname{det} A^{-1}=1 $</p>
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<p>$ \operatorname{det}  A^{-1}=\frac{1}{\operatorname{det} A} $</p>
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<p>$ \operatorname{det}(\operatorname{adj} A)=(\operatorname{det} A)^{n-1} A \rightarrow \text { nxn matrix. }$</p>
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<footer style = "font-size: 18px">Adjoint of a matrix $\rightarrow$ Adjoint of a matrix contd... $\rightarrow$ <span style = "color:blue"> Determinant of adjoint of a matrix </span> $\rightarrow$ Problem-1 $\rightarrow$ Solution </footer>

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<p>Let $M$ be $3 \times 3$ matrix satisfying</p>
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<p>M$\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right], $</p>
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<p>$M\left[\begin{array}{c}1 \\ -1 \\ 0 \end{array}\right]=\left[\begin{array}{c} 1 \\ 1 \\ -1 \end{array}\right], $</p>
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<p>$M\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 12 \end{array}\right]$</p>
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<p>Then, what will be the sum of the diagonal entries of $M$.</p>
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<footer style = "font-size: 18px">Adjoint of a matrix contd... $\rightarrow$ Determinant of adjoint of a matrix $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Solution $\rightarrow$ Solution contd... </footer>

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<p>$ M=\left[\begin{array}{lll} m_{11} & m_n & m_{13} \\ m_{21} & m_{22} & m_{23} \\m_{31} & m_{32} & m_{33}
\end{array}\right] $</p>
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<p>We need to find the value of $m_{11}+m_{22}+m_{33}$</p>
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<p>$M\left[\begin{array}{c}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right] \Rightarrow\left[\begin{array}{c}m_{12} \\ m_{22} \\ m_{32}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right]$</p>
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- $\Rightarrow m_{12}=-1, m_{22}=2, m_{32}=3$

- $M\left[\begin{array}{c}1 \\\ -1 \\\ 0\end{array}\right]=\left[\begin{array}{c}1 \\\ 1 \\\ -1\end{array}\right], \Rightarrow\left[\begin{array}{c}m_{11}-m_{22} \\\ m_{21}-m_{22} \\\ m_{31}-m_{32}\end{array}\right]=\left[\begin{array}{c}1 \\\ 1 \\\ -1\end{array}\right]$

- $\Rightarrow m_{11}-m_{12}=1 \Rightarrow m_{11}=1+m_{12}=1-1=0$

- $m_{11}=0$

- $m_{21}-m_{22}=1 \Rightarrow m_{21}=1+2=3$

- $ m_{31}-m_{32}=-1 $

- $\Rightarrow \quad m_{31}=-1+m_{32}=2 $

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<p>$M\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 12\end{array}\right]$</p>
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<p>$\Rightarrow\left[\begin{array}{l}m_{11}+m_{12}+m_{13} \\ m_{21}+m_{22}+m_{23} \\ m_{31}+m_{32}+m_{33}\end{array}\right]=\left[\begin{array}{c}0 \\ 0 \\ 12\end{array}\right]$</p>
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<p>$ m_{31}+m_{32}+m_{33}=12$</p>
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<p>$m_{33}= 12-m_{31}-m_{32} = 12 - 2 - 3 = 7  $</p>
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<p>$m_{33}=7$</p>
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<p>$ m_{33}=7 m_{11}+m_{22}+m_{33} =0+2+7=9$</p>
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### <primary style="font-size:24px"> Problem-2 </primary>
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- Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non- singular matrices of the form

- $\left[\begin{array}{ccc}1 & a & b \\\ \omega & 1 & c \\\ \omega^2 & \omega & 1\end{array}\right]$

- where each of $a, b, c$ is either $\omega$ or $\omega^2$. Then, what is the cardinality of $S$.

- Solution: Given $S$ in set of all non-singular matrices

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- $\left|\begin{array}{ccc}1 & a & b \\\ \omega & 1 & c \\\ \omega^2 & \omega & 1\end{array}\right| \neq 0$

- $1(1-\omega c)-a\left(\omega-\omega^2 c\right)+b\left(\omega^2-\omega^2\right) \neq 0$

- $ \Rightarrow \quad\left(1-\omega c_2-a w(b-\omega c) \neq 0\right. $

- $ \Rightarrow \quad(1-\omega c)(1-a \omega) \neq 0 $

- $ \Rightarrow \quad 1-\omega c \neq 0 \text { and }(1-a \omega) \neq 0 $

- $\omega^2=1$, a \& c  can take only $\omega$    or

- $\Rightarrow a \neq \omega^2, c \neq \omega^2 {l}a=\omega \\ c=\omega$

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- where as $b$ can be either $\omega$ or  $\omega^2$

- $S= \left[\begin{array}{ccc}1 & \omega & \omega \\\ \omega & 1 & \omega \\\ \omega^2 & \omega & 1\end{array}\right],\left[\begin{array}{ccc}1 & \omega & \omega^2 \\\ \omega & 1 & \omega \\\ \omega^2 & \omega & 1\end{array}\right]$

- $\Rightarrow$ Cardinality of $S=2$

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<footer style = "font-size: 18px">Problem-2 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution contd... </span> $\rightarrow$ Problem-3 $\rightarrow$ Solution </footer>

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<p>Let $P=\left[a_{i j}\right]$ be a $3 \times 3$ matrix and let $Q=[b_ij]$, where</p>
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<p>$b_{i j}=2^{i+j} a_{i j}, 1 \leq i, j \leq 3$</p>
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<p>If the determinant of $P$ is 2 , what will be the value detQ?</p>
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<p>Solution: $Q=\left[\begin{array}{llll}4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 b a_{31} & 32 a_{32}  & 64a_{33}\end{array}\right]$</p>
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### <primary style="font-size:24px"> Solution </primary>
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- $\operatorname{det} Q=\left|\begin{array}{ccc}4 a_{11} & 8 a_{12} & 16 a_{13} \\\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\\ 16 a_{31} & 32 a_{32} & 64 a_{33}\end{array}\right|$

- $ =4 \times 8 \times 16\left|\begin{array}{llll}a_{11} & 2 a_{12} & 4 a_{13} \\\ a_{21} & 2 a_{22} & 4 a_{23} \\\ a_{31} & 2 a_{32} & 4 a_{33}\end{array}\right| $

- $\begin{array}{l}=2^2 \times 2^3 \times 2^4 \times 2 \times 2^2 \\ =2^{12} \times \operatorname{det} P\end{array}$

- $\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{array}\right| \Rightarrow \operatorname{det} Q=2^{12} \times 2=2^{13}$

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<footer style = "font-size: 18px">Solution contd... $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-4 $\rightarrow$ Solution </footer>

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### <primary style="font-size:24px"> Problem-4 </primary>
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-  let $P$ be a $3 \times 3$ matrix such that $\quad P^{\top}=2 P+I$, where $I$ is the $3 \times 3$ identity matrix. Consider a column matrix $x=\left[\begin{array}{l}x \\\ y \\\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\\ 0 \\\ 0\end{array}\right]$

- Then, which of the following is true

- i) $P X=\left[\begin{array}{l}0 \\\ 0 \\\ 0\end{array}\right] \quad \quad$ ii) $P X=X$

- iii) $P_X=2 x$ $\quad \quad$ iv) $p x=-x$

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### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ P=2 P^{\top}+I $  $\leftarrow$(I)

- $  \left(P^{\top}\right)^{\top}=(2 P+I)^{\top} $

- $ P=2 P^{\top}+I$ $\leftarrow$(II)

- $P^{\top}-P  =2\left(P-P^{\top}\right) $

- $\Rightarrow  3\left(P^{\top}-P\right)=0 \rightarrow \text { Zero matrix } $

- $ \Rightarrow  P^{\top}=P$

- from

- $
\begin{aligned}
& \text { (1) } \quad P=2 P+I \\\\
& \Rightarrow \quad P=-I 
\end{aligned} $

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### <Success style="font-size:25px"> Matrix And Determinant L-1 </Success>
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- i)
- $ P X=\left[\begin{array}{l}0 \\\ 0 \\\0\end{array}\right] $

- $\Rightarrow-I X=\left[\begin{array}{l}0 \\\0 \\\0\end{array}\right] $

- $ \Rightarrow x=\left[\begin{array}{l}0 \\\0 \\\0\end{array}\right] $
- but  x $ \neq\left[\begin{array}{l}0 \\\0 \\\0\end{array}\right] $

- $\Rightarrow$ i) is not true

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