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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 $\theta$.

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 $\left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right]$

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 ${{A}^{-1}}$.

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.

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.

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 $\rho (A)=r$.

Inverse of a Matrix

A non-singular square matrix of order $n$ is invertible if there exists a square matrix B of the same order such that $AB={{I}_{n}}=BA$.   In such a case, we say that the inverse of A is B and we write ${{A}^{-1}}=B$. The inverse of A is given by ${{A}^{-1}}=\frac{1}{|A|}.adj\,A$.   The necessary and sufficient condition for the existence of the inverse of a square matrix A is that $|A|\ne 0$.   Properties of inverse matrix:   If A and B are invertible matrices of the same order, then    (i) ${{({{A}^{-1}})}^{-1}}=A$   (ii) ${{({{A}^{T}})}^{-1}}={{({{A}^{-1}})}^{T}}$   (iii) ${{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}$                  (iv) ${{({{A}^{k}})}^{-1}}={{({{A}^{-1}})}^{k}},k\in N$   [In particular ${{({{A}^{2}})}^{-1}}={{({{A}^{-1}})}^{2}}]$   (v) $adj({{A}^{-1}})={{(adj\,A)}^{-1}}$   (vi) $|{{A}^{-1}}|\,=\frac{1}{|A|}=\,|A{{|}^{-1}}$   (vii) A = diag $({{a}_{1}}{{a}_{2}}...{{a}_{n}})$$\Rightarrow {{A}^{-1}}=diag\,(a_{1}^{-1}a_{2}^{-1}...a_{n}^{-1})$   (viii)  A is symmetric $\Rightarrow$ ${{A}^{-1}}$ is also symmetric.   (ix) A is diagonal, $|A|\ne 0\,\,\Rightarrow {{A}^{-1}}$is also diagonal.   (x) A is a scalar matrix $\Rightarrow$ ${{A}^{-1}}$is also a scalar matrix.   (xi) A is triangular, $|A|\ne 0$$\rightleftharpoons$${{A}^{-1}}$is also triangular.       (xii) Every invertible matrix possesses a unique inverse.   (xiii)  Cancellation law with respect to multiplication   If A is a non-singular matrix i.e., if $|A|\ne 0$, then ${{A}^{-1}}$exists and $AB=AC\Rightarrow {{A}^{-1}}(AB)={{A}^{-1}}(AC)$   $\Rightarrow$ $({{A}^{-1}}A)B=({{A}^{-1}}A)C$   $\Rightarrow$ $IB=IC\Rightarrow B=C$   $\therefore$ $AB=AC\Rightarrow B=C\Leftrightarrow |A|\,\ne 0$.

Let $A=[{{a}_{ij}}]$be a square matrix of order $n$ and let ${{C}_{ij}}$be cofactor of ${{a}_{ij}}$in  A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A   Thus, $adj$$A={{[{{C}_{ij}}]}^{T}}\Rightarrow {{(adj\,A)}_{ij}}={{C}_{ji}}=$cofactor of ${{a}_{ji}}$in A.     If $A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\end{matrix} \right],$ then  $adj\,A={{\left[ \begin{matrix} {{C}_{11}} & {{C}_{12}} & {{C}_{13}} \\ {{C}_{21}} & {{C}_{22}} & {{C}_{23}} \\ {{C}_{31}} & {{C}_{32}} & {{C}_{33}} \\\end{matrix} \right]}^{T}}=\left[ \begin{matrix} {{C}_{11}} & {{C}_{21}} & {{C}_{31}} \\ {{C}_{12}} & {{C}_{22}} & {{C}_{32}} \\ {{C}_{13}} & {{C}_{23}} & {{C}_{33}} \\\end{matrix} \right];$ where ${{C}_{ij}}$denotes the cofactor of ${{a}_{ij}}$in A.   Example : $A=\left[ \begin{matrix} p & q \\r & s \\\end{matrix} \right],\,{{C}_{11}}=s,\,{{C}_{12}}=-r,\,{{C}_{21}}=-q,\,{{C}_{22}}=p$ $\therefore adj\,A={{\left[ \begin{matrix} s & -r \\ -q & p \\\end{matrix} \right]}^{T}}=\left[ \begin{matrix} s & -q \\ -r & p \\\end{matrix} \right]$   Properties of adjoint matrix : If A, B are square matrices of order $n$ and ${{I}_{n}}$is corresponding unit matrix, then   (i) $A(adj\,A)=|A|{{I}_{n}}=(adj\,A)A$   (Thus A (adj A) is always a scalar matrix)   (ii) $|adj\,A|=|A{{|}^{n-1}}$                                 (iii) $adj\,(adj\,A)=|A{{|}^{n-2}}A$   (iv) $|adj\,(adj\,A)|\,=\,|A{{|}^{{{(n-1)}^{2}}}}$                  (v) $adj\,({{A}^{T}})={{(adj\,A)}^{T}}$   (vi) $adj\,(AB)=(adj\,B)(adj\,A)$           (vii) $adj({{A}^{m}})={{(adj\,A)}^{m}},m\in N$   (viii) $adj(kA)={{k}^{n-1}}(adj\,A),k\in R$   (ix) $adj\,({{I}_{n}})={{I}_{n}}$                               (x) $adj\,(O)=O$   (xi) A is symmetric $\Rightarrow$ adj A is also symmetric.   (xii) A is diagonal $\Rightarrow$ adj A is also diagonal.   (xiii) A is triangular $\Rightarrow$ adj A is also triangular.   (xiv) A is singular $\Rightarrow$ $|adj\,\,A|=0$

Special Types of Matrices

(1) Symmetric matrix : A square matrix $A=[{{a}_{ij}}]$is called symmetric matrix if ${{a}_{ij}}={{a}_{ji}}$for all i, j or ${{A}^{T}}=A$.   Example : $\left[ \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right]$   (2) Skew-symmetric matrix : A square matrix $A=[{{a}_{ij}}]$is called skew- symmetric matrix if ${{a}_{ij}}=-{{a}_{ji}}$for all i, j or ${{A}^{T}}=-A$.   Example : $\left[ \begin{matrix} 0 & h & g \\ -h & 0 & f \\ -g & -f & 0 \\ \end{matrix} \right]$   All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.   ${{a}_{ij}}=-{{a}_{ij}}\Rightarrow {{a}_{ij}}=0$   Properties of symmetric and skew-symmetric matrices   (i) If A is a square matrix, then $A+{{A}^{T}},A{{A}^{T}},{{A}^{T}}A$ are symmetric matrices, while $A-{{A}^{T}}$is skew- symmetric matrix.   (ii) If A is a symmetric matrix, then$-A,KA,{{A}^{T}},{{A}^{n}},{{A}^{-1}},{{B}^{T}}AB$ are also symmetric matrices, where $n\in N$, $K\in R$ and B is a square matrix of order that of A.   (iii) If A is a skew-symmetric matrix, then   (a) ${{A}^{2n}}$is a symmetric matrix for $n\in N$.   (b) ${{A}^{2n+1}}$is a skew-symmetric matrix for $n\in N$.   (c) kA is also skew-symmetric matrix, where $k\in R$.   (d)  ${{B}^{T}}AB$ is also skew- symmetric matrix where B is a square matrix of order that of A.   (iv) If A, B are two symmetric matrices, then   (a)  $A\pm B,\,\,AB+BA$ are also symmetric matrices,   (b)  $AB-BA$is a skew- symmetric matrix,   (c)   AB is a symmetric matrix, when $AB=BA$.   (v) If A, B  are two skew-symmetric matrices, then   (a) $A\pm B,\,\,AB-BA$ are skew-symmetric matrices,   (b) $AB+BA$is a symmetric matrix.   (vi) If A a skew-symmetric matrix and C is a column matrix, then ${{C}^{T}}$AC is a zero matrix.   (vii) Every square matrix A can unequally be expressed as sum of a symmetric and skew-symmetric matrix   i.e., $A=\left[ \frac{1}{2}(A+{{A}^{T}}) \right]+\left[ \frac{1}{2}(A-{{A}^{T}}) \right]$.   (3) Singular and Non-singular matrix : Any square matrix A is said to be non-singular if $|A|\ne 0,$and a square matrix A is said to be singular if $|A|\,=0$. Here $|A|$(or det(A) or simply det  $|A|$ means corresponding determinant of square matrix A.   Example : $A=\left[ \begin{matrix} 2 & 3 \\ 4 & 5 \\ \end{matrix} \right]$ then$|A|\,=\left| \,\begin{matrix} 2 & 3 \\ 4 & 5 \\\end{matrix}\, \right|=10-12=-2\Rightarrow A$ is a non-singular matrix.   (4) Hermitian and Skew-hermitian matrix : A square matrix $A=[{{a}_{ij}}]$ is said to be hermitian matrix if   ${{a}_{ij}}={{\bar{a}}_{ji}}\,;\,\,\forall i,j\,\,i.e.,\,A={{A}^{\theta }}$.   Example : $\left[ \begin{matrix} a & b+ic \\ b-ic & d \\ \end{matrix} \right]\,,\,\,\left[ \begin{matrix} 3 & 3-4i & 5+2i \\ 3+4i & 5 & -2+i \\ 5-2i & -2-i & 2 \\ \end{matrix} \right]$   are Hermitian matrices. If A is a Hermitian matrix then ${{a}_{ii}}={{\bar{a}}_{ii}}\,\,\Rightarrow$${{a}_{ii}}$ is real $\forall i,$ thus every diagonal element of a Hermitian matrix must be real.   A square matrix, $A=\,\,|{{a}_{jj}}|$ is said to be a Skew-Hermitian if ${{a}_{ij}}=-{{\bar{a}}_{ji}}.\,\forall i,\,j\,i.e.\,{{A}^{\theta }}=-A$. If A is a more...

Transpose of a Matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by ${{A}^{T}}$or ${A}'$.   From the definition it is obvious that if order of A is $m\times n,$ then order of ${{A}^{T}}$is $n\times m$.   Example:   Transpose of matrix ${{\left[ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ \end{matrix} \right]}_{2\times 3}}$ is $\text{ }{{\left[ \begin{matrix} {{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & {{b}_{2}} \\ {{a}_{3}} & {{b}_{3}} \\ \end{matrix} \right]}_{3\times 2}}$   Properties of transpose : Let A and B be two matrices then,   (i)  ${{({{A}^{T}})}^{T}}=A$   (ii)  ${{(A+B)}^{T}}={{A}^{T}}+{{B}^{T}},A$and B being of the same order   (iii)  ${{(kA)}^{T}}=k{{A}^{T}},k$ be any scalar (real or complex)   (iv) ${{(AB)}^{T}}={{B}^{T}}{{A}^{T}},A$ and B being conformable for the product AB   (v) ${{({{A}_{1}}{{A}_{2}}{{A}_{3}}.....{{A}_{n-1}}{{A}_{n}})}^{T}}={{A}_{n}}^{T}{{A}_{n-1}}^{T}.......{{A}_{3}}^{T}{{A}_{2}}^{T}{{A}_{1}}^{T}$   (vi) ${{I}^{T}}=I$

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