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(1) Row matrix : A matrix is said to be a row matrix or row vector if it has only one row and any number of columns.     Example :  [5  0  3] is a row matrix of order \[1\times 3\] and [2] is a row matrix of order \[1\times 1\].   (2) Column matrix : A matrix is said to be a column matrix or column vector if it has only one column and any number of rows.   Example : \[\left[ \begin{align} & \,\,\,2 \\  & \,\,\,3 \\  & -6 \\  \end{align} \right]\] is a column matrix of order \[3\times 1\] and [2] is a column matrix of order \[1\times 1\]. Observe that [2] is both a row matrix as well as a column matrix.   (3) Singleton matrix : If in a matrix there is only one element then it is called singleton matrix.   Thus, \[A={{[{{a}_{ij}}]}_{m\times n}}\]is a singleton matrix, if \[m=n=1\]   Example : \[[2],\text{ }[3],\text{ }[a],\text{ }[3]\] are singleton matrices.   (4) Null or zero matrix : If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by \[O\]. Thus \[A={{[{{a}_{ij}}]}_{m\times n}}\]is a zero matrix if \[{{a}_{ij}}=0\]for all \[i\] and \[j\].   Example : \[[0],\left[ \begin{matrix} 0 & 0  \\ 0 & 0  \\ \end{matrix} \right],\left[ \begin{matrix} 0 & 0 & 0  \\ 0 & 0 & 0  \\ \end{matrix} \right],[0\,\,0]\] are all zero matrices, but of different orders.   (5) Square matrix : If number of rows and number of columns in a matrix are equal, then it is called a square matrix.   Thus \[A={{[{{a}_{ij}}]}_{m\times n}}\]is a square matrix if \[m=n\].   Example : \[\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}}  \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}}  \\ \end{matrix} \right]\]is a square matrix of order \[3\times 3\].   (i) If \[m\ne n\]then matrix is called a rectangular matrix.   (ii) The elements of a square matrix A for which \[i=j,i.e.\,\,{{a}_{11}},\] \[{{a}_{22}},{{a}_{33}},....{{a}_{nn}}\]are called diagonal elements and the line joining these elements is called the principal diagonal or leading diagonal of matrix A.   (6) Diagonal matrix : If all elements except the principal diagonal in a square matrix are zero, it is called a diagonal matrix. Thus a square matrix \[A=[{{a}_{ij}}]\] is a diagonal matrix if \[\Delta \]when \[\Delta =0\].   Example : \[\left[ \begin{matrix} 2 & 0 & 0  \\ 0 & 3 & 0  \\ 0 & 0 & 4  \\ \end{matrix} \right]\]is a diagonal matrix of order \[3\times 3\], which can be denoted by diag [2, 3, 4].   (7) Identity matrix : A square matrix in which elements in the main diagonal are all '1' and rest are all zero is called an identity matrix or unit matrix. Thus, the square matrix \[A=[{{a}_{ij}}]\]is an identity matrix, if \[{{a}_{ij}}=\left\{ \begin{align} & 1,\,\,\text{if}\,\,\,i=j \\  & 0,\,\,\text{if}\,\,i\ne j \\  \end{align} \right.\]   We denote the identity matrix of order \[n\] by \[{{I}_{n}}\].   Example : [1], more...

 The sum of diagonal elements of a square matrix. A is called the trace of matrix A, which is denoted by tr A.   \[tr\,\,A=\sum\limits_{i=1}^{n}{{{a}_{ii}}={{a}_{11}}+{{a}_{22}}+...{{a}_{nn}}}\]   Properties of trace of a matrix   Let \[{{C}_{11}},\,{{C}_{12}},\,{{C}_{13}}\]and \[B={{[{{b}_{ij}}]}_{n\times n}}\]and \[\lambda \]be a scalar   (i) \[tr(\lambda A)=\lambda \,tr(A)\]                     (ii) \[tr(A-B)=tr(A)-\,tr\,(B)\]   (iii) \[tr(AB)=tr(BA)\]                               (iv) \[tr\,(A)\,=tr\,(A')\] or \[t{{r}_{{}}}({{A}^{T}})\]   (v) \[tr\,({{I}_{n}})=n\]   (vi) \[tr\,(0)\,=0\]   (vii) \[tr\,(AB)\ne tr\,A\,.\,tr\,B\]  

If \[A={{[{{a}_{ij}}]}_{m\times n}}\]and \[B={{[{{b}_{ij}}]}_{m\times n}}\]are two matrices of the same order then their sum \[A+B\] is a matrix whose each element is the sum of corresponding elements i.e., \[A+B={{[{{a}_{ij}}+{{b}_{ij}}]}_{m\times n}}\].   Similarly, their subtraction \[A-B\] is defined as   \[A-B={{[{{a}_{ij}}-{{b}_{ij}}]}_{m\times n}}\]   Matrix addition and subtraction can be possible only when matrices are of the same order.   Properties of matrix addition : If A, B and C are matrices of same order, then   (i) \[A+B=B+A\]                    (Commutative law)                    (ii) \[(A+B)+C=A+(B+C)\]    (Associative law)   (iii) \[A+O=O+A=A,\]where O is zero matrix which is additive identity of the matrix.   (iv) \[A+(-A)=0=(-A)+A\], where \[(-A)\] is obtained by changing the sign of every element of A, which is additive inverse of the matrix.   (v) \[\left. \begin{align}  & A+B=A+C \\  & B+A=C+A \\  \end{align} \right\}\Rightarrow B=C\]          (Cancellation law)

 Let \[A={{[{{a}_{ij}}]}_{m\times n}}\]be a matrix and k be a number, then the matrix which is obtained by multiplying every element of A by k is called scalar multiplication of A by k and it is denoted by kA.   Thus, if \[A={{[{{a}_{ij}}]}_{m\times n}}\], then \[kA=Ak={{[k{{a}_{ij}}]}_{m\times n}}\].   Properties of scalar multiplication   If A, B are matrices of the same order and \[\lambda ,\,\mu \] are any two scalars then   (i) \[\lambda (A+B)=\lambda A+\lambda B\]                        (ii) \[(\lambda +\mu )A=\lambda A+\mu A\]   (iii) \[\lambda (\mu A)=(\lambda \mu A)=\mu (\lambda A)\]                 (iv) \[(-\lambda A)=-(\lambda A)=\lambda \,(-A)\]  
  • All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.

Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). Thus, if \[A={{[{{a}_{ij}}]}_{m\times n}}\] and \[B={{[{{b}_{ij}}]}_{n\times p}}\] are two matrices of order \[m\times n\] and \[n\times p\]respectively, then their product AB is of order \[m\times p\]and is defined as \[{{(AB)}_{ij}}=\sum\limits_{r=1}^{n}{{{a}_{ir}}{{b}_{rj}}}\]\[=[{{a}_{i1}}{{a}_{i2}}...{{a}_{in}}]\left[ \begin{align} & \underset{\vdots }{\mathop{\overset{{{b}_{1j}}}{\mathop{{{b}_{2j}}}}\,}}\, \\  & {{b}_{nj}} \\  \end{align} \right]=\] (\[{{i}^{th}}\] row of A)(\[{{j}^{th}}\] column of B)                                                                                                            .....(i)   where \[i=1,\text{ }2,\text{ }...,m\] and \[j=1,\text{ }2,\text{ }...p\]   Now we define the product of a row matrix and a column matrix.   Let \[A=\left[ {{a}_{1}}{{a}_{2}}....{{a}_{n}} \right]\]be a row matrix and \[B=\left[ \begin{matrix} {{b}_{1}}  \\ \underset{\vdots }{\mathop{{{b}_{2}}}}\,  \\ {{b}_{n}}  \\ \end{matrix} \right]\] be a column matrix.   Then \[AB=\left[ {{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+....+{{a}_{n}}{{b}_{n}} \right]\]                             ?..(ii)   Thus, from (i), \[{{(AB)}_{ij}}=\]Sum of the product of elements of \[{{i}^{th}}\] row of A with the corresponding elements of \[{{j}^{th}}\] column of B.   Properties of matrix multiplication   If A, B and C are three matrices such that their product is defined, then   (i) \[AB\ne BA\],           (Generally not commutative)   (ii) \[(AB)C=A(BC)\],          (Associative Law)   (iii) \[IA=A=AI\], where I is identity matrix for matrix multiplication.   (iv) \[A(B+C)=AB+AC\], (Distributive law)   (v)  If \[AB=AC\not{\Rightarrow }B=C\],(Cancellation law is not applicable)   (vi) If \[AB=0,\] it does not mean that \[A=0\] or \[B=0,\] again product of two non zero matrix may be a zero matrix.  

The positive integral powers of a matrix A are defined only when A is a square matrix.   Also then \[{{A}^{2}}=A.A\], \[{{A}^{3}}=A.A.A={{A}^{2}}A\].    Also for any positive integers \[m\] and \[n,\]   (i) \[{{A}^{m}}{{A}^{n}}={{A}^{m+n}}\]    (ii) \[{{({{A}^{m}})}^{n}}={{A}^{mn}}={{({{A}^{n}})}^{m}}\]   (iii) \[{{I}^{n}}=I,{{I}^{m}}=I\]                                                   (iv) \[{{A}^{0}}={{I}_{n}}\], where A is a square matrix of order \[n\].  

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) \[{{I}^{T}}=I\]

(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...

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

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 \[AB=AC\Rightarrow {{A}^{-1}}(AB)={{A}^{-1}}(AC)\]   \[\Rightarrow \] \[({{A}^{-1}}A)B=({{A}^{-1}}A)C\]   \[\Rightarrow \] \[IB=IC\Rightarrow B=C\]   \[\therefore \] \[AB=AC\Rightarrow B=C\Leftrightarrow |A|\,\ne 0\].

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