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

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

when $m=n$, we call $A$ as square matrix.

let $A$ be an $n \times n$ square matrix

$\Rightarrow A^{T}=-A$

$\Rightarrow a_{i j}=-a_{j i} \forall i,j$

$\Rightarrow a_{i j}=-a_{j i}$

$\Rightarrow a_{i j}=0 \forall i$

$\Rightarrow$ diagonal entries are zero.

If $c$ is a scalar then

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

$(A+B)^{\top}=A^{\top}+B^{\top}$

$(A B)^{\top}=B^{\top} A^{\top}$

$n=2, \quad A =\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]$

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

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

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

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

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

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

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, $\operatorname{det} A=0$

v)

$\quad\left|\begin{array}{ccc}k a_{11} & k a_{12} & k a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=k\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$

vi)

$\begin{aligned}\left|\begin{array}{ccc}a_{11} & a_{12}+x & a_{13} \\ a_{21} & a_{22}+y & a_{23} \\ a_{31} & a_{32}+z & a_{33}\end{array}\right| & =\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right| +\left|\begin{array}{lll}a_{12} & x & a_{13} \\ a_{21} & y & a_{23} \\ a_{31} & z & a_{33}\end{array}\right|\end{aligned}$

vii) $\operatorname{det}(A B)=\operatorname{det} A \cdot \operatorname{det} B$

viii) $\operatorname{det}(c A)=c^n \operatorname{det} A$

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 $\operatorname{det} A \neq 0$

We call an invertible matrix as non-singular matrix.

$\operatorname{adj} A=$ transpose of co-factor matrix.

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

$A_{i j}=(-1)^{i+j} \times$ 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.

$\operatorname{det} A^{-1}=\frac{1}{\operatorname{det} A}$

$A A^{-1}=I$

$\operatorname{det}\left(A A^{-1}\right)\operatorname{det} I =1$

$\operatorname{det} A \cdot \operatorname{det} A^{-1}=1$

$\operatorname{det} A^{-1}=\frac{1}{\operatorname{det} A}$

$\operatorname{det}(\operatorname{adj} A)=(\operatorname{det} A)^{n-1} A \rightarrow \text { nxn matrix. }$

Let $M$ be $3 \times 3$ matrix satisfying

M$\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right],$

$M\left[\begin{array}{c}1 \\ -1 \\ 0 \end{array}\right]=\left[\begin{array}{c} 1 \\ 1 \\ -1 \end{array}\right],$

$M\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 12 \end{array}\right]$

Then, what will be the sum of the diagonal entries of $M$.

$M=\left[\begin{array}{lll} m_{11} & m_n & m_{13} \\ m_{21} & m_{22} & m_{23} \\m_{31} & m_{32} & m_{33} \end{array}\right]$

We need to find the value of $m_{11}+m_{22}+m_{33}$

$M\left[\begin{array}{c}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right] \Rightarrow\left[\begin{array}{c}m_{12} \\ m_{22} \\ m_{32}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right]$

$\Rightarrow m_{12}=-1, m_{22}=2, m_{32}=3$

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

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

$m_{11}=0$

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

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

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

$M\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 12\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}m_{11}+m_{12}+m_{13} \\ m_{21}+m_{22}+m_{23} \\ m_{31}+m_{32}+m_{33}\end{array}\right]=\left[\begin{array}{c}0 \\ 0 \\ 12\end{array}\right]$

$m_{31}+m_{32}+m_{33}=12$

$m_{33}= 12-m_{31}-m_{32} = 12 - 2 - 3 = 7$

$m_{33}=7$

$m_{33}=7 m_{11}+m_{22}+m_{33} =0+2+7=9$

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

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

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

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

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

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

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

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

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 $S=2$

Let $P=\left[a_{i j}\right]$ be a $3 \times 3$ matrix and let $Q=[b_ij]$, where

$b_{i j}=2^{i+j} a_{i j}, 1 \leq i, j \leq 3$

If the determinant of $P$ is 2 , what will be the value detQ?

Solution: $Q=\left[\begin{array}{llll}4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 b a_{31} & 32 a_{32} & 64a_{33}\end{array}\right]$

$\operatorname{det} Q=\left|\begin{array}{ccc}4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 a_{31} & 32 a_{32} & 64 a_{33}\end{array}\right|$

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

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

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

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) $p x=-x$

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

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

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