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Rank of Matrix

Definition : Let A be a $m\times n$ matrix. If we retain any $r$ rows and $r$ columns of A we shall have a square sub-matrix of order $r$. The determinant of the square sub-matrix of order $r$ is called a minor of A order $r$. Consider any matrix A which is of the order of $3\times 4$ say, $A=\left| \begin{matrix} 1 & 3 & 4 & 5 \\ 1 & 2 & 6 & 7 \\ 1 & 5 & 0 & 1 \\ \end{matrix} \right|$. It is $3\times 4$ matrix so we can have minors of order 3, 2 or 1. Taking any three rows and three columns minor of order three. Hence minor of order $3=\left| \,\begin{matrix} 1 & 3 & 4 \\ 1 & 2 & 6 \\ 1 & 5 & 0 \\ \end{matrix}\, \right|=0$   Making two zeros and expanding above minor is zero. Similarly we can consider any other minor of order 3 and it can be shown to be zero. Minor of order 2 is obtained by taking any two rows and any two columns.   Minor of order ${{D}_{3}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{d}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{d}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{d}_{3}} \\ \end{matrix}\, \right|$. Minor of order 1 is every element of the matrix.   Rank of a matrix: The rank of a given matrix A is said to be $r$ if     (1) Every minor of A of order $r+1$ is zero.   (2) There is at least one minor of A of order $r$ which does not vanish. Here we can also say that the rank of a matrix A is said to be $r$, if (i) Every square submatrix of order $r+1$ is singular.   (ii) There is at least one square submatrix of order $r$ which is non-singular.   The rank $r$  of matrix A is written as $\rho (A)=r$.

Echelon Form of a Matrix

A matrix A is said to be in Echelon form if either A is the null matrix or A satisfies the following conditions:   (1) Every non- zero row in A precedes every zero row.   (2) The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.   If can be easily proved that the rank of a matrix in Echelon form is equal to the number of non-zero row of the matrix.   Rank of a matrix in Echelon form : The rank of a matrix in Echelon form is equal to the number of non-zero rows in that matrix.

Homogeneous and Non-homogeneous Systems of Linear Equations

A system of equations $AX=B$ is called a homogeneous system if $B=O$. If $B\ne O$, it is called a non-homogeneous system of equations. e.g., $2x+5y=0$ $3x-2y=0$   is a homogeneous system of linear equations whereas the system of equations given by e.g., $2x+3y=5$ $x+y=2$   is a non-homogeneous system of linear equations.   (1) Solution of Non-homogeneous system of linear equations   (i) Matrix method : If $AX=B$, then $X={{A}^{-1}}B$ gives a unique solution, provided A is non-singular.   But if A is a singular matrix i.e.,  if $|A|=0$, then the system of equation $AX=B$ may be consistent with infinitely many solutions or it may be inconsistent.   (ii) Rank method for solution of Non-Homogeneous system $AX=B$   (a) Write down A, B   (b) Write the augmented matrix $[A:B]$   (c) Reduce the augmented matrix to Echelon form by using elementary row operations.   (d) Find the number of non-zero rows in A and $[A:B]$ to find the ranks of A and $[A:B]$ respectively.   (e) If $\rho (A)\ne \rho (A:B),$ then the system is inconsistent.   (f) $\rho (A)=\rho (A:B)=$ the number of unknowns, then the system has a unique solution.   If $\rho (A)=\rho (A:B)<$ number of unknowns, then the system has an infinite number of solutions.   (2) Solutions of a homogeneous system of linear equations : Let $AX=O$ be a homogeneous system of 3 linear equations in 3 unknowns.   (a) Write the given system of equations in the form $AX=O$ and write A.   (b) Find $|A|$.   (c) If $|A|\ne 0$, then the system is consistent and $x=y=z=0$ is the unique solution.   (d)  If $|A|=0$, then the systems of equations has infinitely many solutions. In order to find that put $z=K$ (any real number) and solve any two equations for $x$ and $y$ so obtained with $z=K$ give a solution of the given system of equations.

Consistency of a System of Linear Equation

In system of linear equations $AX=B,\,A={{({{a}_{ij}})}_{n\times n}}$ is said to be   (i) Consistent (with unique solution) if $|A|\ne 0$.   i.e., if $A$ is non-singular matrix.   (ii) Inconsistent (It has no solution) if $|A|=0$ and $(adjA)\,B$ is a non-null matrix.   (iii) Consistent (with infinitely $m$ any solutions) if $|A|\,=\,0$ and $(adj\,A)\,B$ is a null matrix.

Cayley-Hamilton Theorem

Every matrix satisfies its characteristic equation e.g. let A be a square matrix then $|A-xI|=0$is the characteristics equation of A. If ${{x}^{3}}-4{{x}^{2}}-5x-7=0$ is the characteristic equation for A, then ${{A}^{3}}-4{{A}^{2}}+5A-7I=0$.   Roots of characteristic equation for A are called Eigen values of A or characteristic roots of A or latent roots of A.   If $\lambda$ is characteristic root of A, then ${{\lambda }^{-1}}$is characteristic root of ${{A}^{-1}}$.

Geometrical Transformations

(1) Reflexion in the x-axis: If $P'\,\,(x',y')$is the reflexion of the point $P(x,y)$on the x-axis, then the matrix $\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \\\end{matrix} \right]$ describes the reflexion of a point $P(x,y)$in the x-axis.   (2) Reflexion in the y-axis    Here the matrix is $\left[ \begin{matrix} -1 & 0 \\ 0 & 1 \\\end{matrix} \right]$   (3) Reflexion through the origin   Here the matrix is $\left[ \begin{matrix} -1 & 0 \\ 0 & -1 \\ \end{matrix} \right]$   (4) Reflexion in the line  $\mathbf{y=x}$   Here the matrix is $\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]$   (5) Reflexion in the line $\mathbf{y=}-\mathbf{x}$   Here the matrix is $\left[ \begin{matrix} \,\,0 & -1 \\ -1 & \,\,0 \\ \end{matrix} \right]$   (6) Reflexion in $y=x\,\mathbf{tan\theta }$   Here matrix is $\left[ \begin{matrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \\ \end{matrix} \right]$   (7) Rotation through an angle $\mathbf{\theta }$   Here matrix is $\left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right]$

Matrices of Rotation of Axes

We know that if $x$ and $y$ axis are rotated through an angle $\theta$ about the origin the new coordinates are given by   $x=X\,\cos \theta -Y\sin \theta$ and $y=X\sin \theta +Y\cos \theta$   $\Rightarrow \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]=\left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right]\,\left[ \begin{matrix} X \\ Y \\ \end{matrix} \right]\Rightarrow \left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right]$   is the matrix of rotation through an angle $\theta$.

Domain and Range of a Trigonometrical Function

If $f:X\to Y$ is a function, defined on the set $X,$ then the domain of the function $f,$ written as Domf is the set of all independent variables $x,$ for which the image $f(x)$ is well defined element of $Y,$ called the co-domain of $f$.   Range of $f:X\to Y$is the set of all images ${{72}^{o}}$ which belongs to $Y,$ i.e., Range ${{67.5}^{o}}$$\{f(x)\in Y:x\in X\}\,\subseteq Y$.   The domain and range of trigonometrical functions are tabulated as follows :
 Trigonometrical Function Domain Range $\sin x$ $R$ $-1\le \sin x\le 1$ $\cos x$ $R$ $-1\le \cos x\le 1$ $\tan x$ $R-\left\{ (2n+1)\frac{\pi }{2},\,n\in I \right\}$ $R$ $\text{cosec}\,x$ $R-\{n\,\pi ,\,n\in I\}$ $R-\{x:-1 Trigonometrical Ratios or Functions In the right angled triangle \[OMP,$ we have base $=OM=x,$ perpendicular $=PM=y$ and hypotenuse $=OP=r$. We define the following trigonometric ratio which are also known as trigonometric function. $\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenues}}=\frac{y}{r}$   $\frac{2n\pi \pm A}{2}$   $\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{y}{x}$   $\cot \theta =\frac{\text{Base}}{\text{Perpendicular}}=\frac{x}{y}$     $\sec \theta =\frac{\text{Hypotenues}}{\text{Base}}=\frac{r}{x}$   $\text{cosec}\theta =\frac{\text{Hypotenues}}{\text{Perpendicular}}=\frac{r}{y}$   (1) Relation between trigonometric ratios (functions)   (i) $\frac{\sqrt{4-\sqrt{2}-\sqrt{6}}}{2\sqrt{2}}$              (ii) $\tan \theta .\cot \theta =1$   (iii) $\cos \theta .\sec \theta =1$            (iv) $\tan \frac{A}{2}$  (v) $\cot \theta =\frac{\cos \theta }{\sin \theta }$   (2) Fundamental trigonometric identities   (i) ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$                        (ii) $1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta$   (iii) $1+{{\cot }^{2}}\theta =\text{cose}{{\text{c}}^{2}}\theta$   (3) Sign of trigonometrical ratios or functions : Their signs depends on the quadrant in which the terminal side of the angle lies.   In brief: A crude aid to memorise the signs of trigonometrical ratio in different quadrant.  "Add Sugar To Coffee". Algorithm : First determine the sign of the trigonometric function.   If $\theta$ is measured from${X}'OX$ i.e., {(p ± q, 2p – q)} then retain the original name of the function.   If $\theta$ is measured from ${Y}'OY$ i.e.,$\left\{ \frac{\pi }{2}\pm \theta ,\,\frac{3\pi }{2}\pm \theta \right\}$, then change sine to cosine, cosine to sine, tangent to cotangent, cot to tan, sec to cosec and cosec to sec.   (4) Variations in values of trigonometric functions in different quadrants : Let $X'OX$ and $YOY'$ be the coordinate axes. Draw a circle with centre at origin O and radius unity. Let $M(x,y)$ be a point on the circle such that $\angle AOM=\theta$ then $x=\cos \theta$ and $y=\sin \theta$; $-1\le \cos \theta \le$1 and $-1\le \sin \theta \le 1$ for all values of $\theta$.
$\sin \theta \to$ decreases from 1 to 0 $\sin \theta \to$ increases from 0 to 1
$\cos \theta \to$ decreases from 0 to - 1 more...

Trigonometrical Ratios of Allied Angles

Two angles are said to be allied when their sum or difference is either zero or a multiple of ${{90}^{o}}$.
Allied angles $\to$ $\sin \theta$ $cos\theta$ $tan\theta$
Trigo. Ratio
$\downarrow \,\,(-\theta )$ $-\sin \theta$ $cos\theta$ $-tan\theta$
$(90-\theta )$ or $\left( \frac{\pi }{2}-\theta \right)$ $cos\theta$ $\sin \theta$ $\cot \,\theta$
$(90-\theta )$ or $\left( \frac{\pi }{2}-\theta \right)$ $\cos \theta$ $-\,\sin \theta$ $-\cot \,\theta$
$(180-\theta )$ or$(\pi -\theta )$ $\sin \theta$ $-\,\cos \theta$ more...

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