\[\LARGE{\color {purple} {MATRICES\ and\ DETERMINANTS}}\]
\[\color {green} {Determinant\ of\ Second\ order:}\ \hspace{20cm}\]
\[\Delta =\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2 \\
\end{vmatrix}\ = a_1b_2\ -\ a_2b_1\ \hspace{15cm}\]
\[\color {green} {Determinant\ of\ Third\ order:}\ \hspace{20cm}\]
\[\begin{vmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3 \\
\end{vmatrix}\ \hspace{20cm}\]
\[\Delta =a_1\begin{vmatrix}
b_2 & b_3 \\
c_2 & c_3 \\
\end{vmatrix}\ -\ a_2\begin{vmatrix}
b_1 & b_3 \\
c_1 & c_3 \\
\end{vmatrix}\ +\ a_3\begin{vmatrix}
b_1 & b_2 \\
c_1 & c_2 \\
\end{vmatrix}\ \hspace{10cm}\]
\[\Delta =a_1(b_2c_3\ -\ b_3c_2)\ – a_2 (b_1c_3\ -\ b_3c_1) + a_3(b_1c_2\ -\ b_2c_1)\
\hspace{10cm}\]
\[\color {green} {Singular\ and\ Non-Singular\ Matrix}:\ \hspace{20cm}\]
A square matrix A is called a singular matrix
\[if\ \begin{vmatrix} A \\ \end{vmatrix}\ = 0\ and\ non\ –\ singular\ matrix\ if\ \begin{vmatrix} A \\ \end{vmatrix}\ \neq {0}\ \hspace{10cm}\]
\[\LARGE{\color {purple} {APPLICATION\ OF\ MATRICES\ and\ DETERMINANTS}}\]
\[\color {purple} {Minor\ of\ an\ element\ of\ a\ Matrix}\ \hspace{20cm}\]
\[Minor\ of\ an\ element\ is\ a\ determinant\ obtained\ by\ deleting\ the\ row\ and\ column\ in\ which\ the\ element\ occurs\]
\[\color {purple} {Cofactor\ of\ an\ element of\ a\ Matrix}\ \hspace{20cm}\]
\[Cofactor\ of\ an\ element\ is\ a\ signed\ minor\ of\ that\ element\]
\[\therefore\ cofactor\ of\ a_{ij} = (-1)^{i\ +\ j}\ minor\ of\ a_{ij}\]
\[\color {purple}{Method\ for\ to\ find\ adjoint\ of\ Matrix\ of\ order\ 3\ (order 2)}\ \hspace{20cm}\]
\[i)\ A\ is\ square\ Matrix\ of\ order\ 3\ (order 2)\ \hspace{5cm}\]
\[ii)\ Find\ the\ co-factor\ of\ all\ the\ elements\ of\ det\ A\ \hspace{8cm}\]
\[iii)\ Form\ the\ matrix\ by\ replacing\ all\ the\ elements\ of\ A\ by\ the\ corresponding\ cofactor\ in\ \begin{vmatrix}
A \\
\end{vmatrix}\]
\[iv)\ Then\ take\ the\ Transpose\ of\ that\ matrix,\ then\ we\ get\ adj. A.\ \hspace{8cm}\]
\[\color {purple} {Inverse\ of\ Matrix}\ \hspace{20cm}\]
\[\color {green} {\boxed {A^{-1} = \frac{1}{\begin{vmatrix} A \\ \end{vmatrix}}\ adj.\ A}}\]
\[\color {purple} {Rank\ of\ Matrix:}\ \hspace{20cm}\]
\[Let\ A\ be\ any\ m×n\ matrix.\ The\ order\ of\ the\ largest\ square\ sub\ matrix\ of\ A\ whose\ determinant\]\[ has\ a\ non\ -\ zero\ value\ is\ known\ as\ the\ rank\ of\ the\ matrix\ A\]\[and\ is\ denoted\ by\ \rho(A)\]
\[\color {purple} {SOLUTION\ OF\ SIMULTANEOUS\ EQUATIONS\ USING\ CRAMERS\ RULE}\ \hspace{20cm}\]
\[a_1x\ + b_1y\ +\ c_1z\ = d_1\ —– (1)\ \hspace{20cm}\]
\[a_2x\ + b_2y\ +\ c_2z\ = d_2\ \hspace{20cm}\]
\[a_3x\ + b_3y\ +\ c_3z\ = d_3\ \hspace{20cm}\]
\[\color {black}{Solution:}\ to\ find\ x,\ y,\ z\ \hspace{20cm}\]
\[Step\ 1:\ \Delta = \begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3 \\
\end{vmatrix}\ \hspace{20cm}\]
\[Step\ 2:\ \Delta_x = \begin{vmatrix}
d_1 & b_1 & c_1 \\
d_2 & b_2 & c_2 \\
d_3 & b_3 & c_3 \\
\end{vmatrix}\ \hspace{20cm}\]
\[Step\ 3:\ \Delta_y= \begin{vmatrix}
a_1 & d_1 & c_1 \\
a_2 & d_2 & c_2 \\
a_3 & d_3 & c_3 \\
\end{vmatrix}\ \hspace{20cm}\]
\[Step\ 4:\ \Delta_z = \begin{vmatrix}
a_1 & b_1 & d_1 \\
a_2 & b_2 & d_2 \\
a_3 & b_3 & d_3 \\
\end{vmatrix}\ \hspace{20cm}\]
\[Solution\ is\ x=\ \frac{\Delta_x}{\Delta}.\ y=\ \frac{\Delta_y}{\Delta},\ z=\ \frac{\Delta_z}{\Delta}\ \hspace{15cm}\]
\[\LARGE{\color {purple} {BINOMIAL\ THEOREM}}\]
\[\color {royalblue} {Binomial\ theorem\ for\ a\ positive\ integral\ index}:\ \hspace{20cm}\]
\[If\ ‘n’\ is\ any\ positive\ integer,\ then\ \hspace{15cm}\]
\[(X\ +\ a)^n\ =\ X^n\ +\ nc_1 X^{n-1}a\ +\ \ nc_2 X^{n-2}a^2\ +\ …………………..\ +\ \ nc_r X^{n-r}a^r\ +\ …….\ a^n\]
\[\color {royalblue} {General\ term\ of\ the\ expansion\ of\ (X\ +\ a)^n}:\ \hspace{20cm}\]
\[T_{r + 1} = nC_rx^{n-r} a^r.\ \hspace{15cm}\]
\[\color {royalblue} {To\ find\ the\ Middle\ term\ of\ the\ expansion\ of\ (X\ +\ a)^n}:\ \hspace{20cm}\]
\[\color {brown} {cases}:\ \hspace{18cm}\]
\[1.\ If\ n\ is\ an\ even\ number,\ there\ is\ one\ middle\ term\ =\ (\frac{n+2}{2})^{th}\ term\]
\[2.\ If\ n\ is\ odd\ number,\ there\ are\ two\ middle\ terms\ (\frac{n+1}{2})^{th}\ and\ \ (\frac{n+1}{2})^{th}\ term\]
\[\color {royalblue} {Binomial\ theorem\ for\ rational\ index}:\ \hspace{20cm}\]
\[If\ ‘n’\ is\ any\ rational\ number\ then\ \hspace{15cm}\]
\[(1\ +\ x)^n\ =\ 1\ +\ nx\ +\ \frac{n(n-1)}{1.\ 2}\ x^2\ +\ \frac{n(n-1)(n-2)}{1.\ 2.\ 3}\ x^3\ +\ …….\]
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