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 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 \[\theta \].  

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

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 \[{{A}^{-1}}\].

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.

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.

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 \[\rho (A)=r\].

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 \[\Rightarrow \] \[|adj\,\,A|=0\]

(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...

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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