\displaystyle 1.\ \text{A set of }mn\text{ numbers (real or imaginary) arranged in the form of a}
\displaystyle \text{rectangular array of }m\text{ rows and }n\text{ columns is called an }m\times n
\displaystyle \text{matrix.}

\displaystyle 2.\ \text{A matrix having only one row is called a row matrix.}

\displaystyle 3.\ \text{A matrix having only one column is called a column matrix.}

\displaystyle 4.\ \text{A matrix in which the number of rows is equal to the number of columns,}
\displaystyle \text{say }n,\text{ is called a square matrix of order }n.

\displaystyle 5.\ \text{The elements }a_{ij}\text{ of a square matrix }A=[a_{ij}]_{n\times n}\text{ for which}
\displaystyle i=j,\text{ i.e. the elements }a_{11},a_{22},\ldots,a_{nn}\text{ are called the diagonal}
\displaystyle \text{elements and the line along which they lie is called the principal}
\displaystyle \text{diagonal or leading diagonal.}

\displaystyle 6.\ \text{A square matrix }A=[a_{ij}]_{n\times n}\text{ is called a diagonal matrix if}
\displaystyle \text{all the elements, except those in the leading diagonal, are zero i.e.}
\displaystyle a_{ij}=0\text{ for }i\neq j.

\displaystyle 7.\ \text{A square matrix }A=[a_{ij}]_{n\times n}\text{ is called a scalar matrix, if}
\displaystyle (i)\ a_{ij}=0\text{ for all }i\neq j\text{ and, }(ii)\ a_{ii}=c\text{ for all }i,
\displaystyle \text{where }c\neq 0.

\displaystyle 8.\ \text{A square matrix }A=[a_{ij}]_{n\times n}\text{ is called an identity or a}
\displaystyle \text{unit matrix, if }(i)\ a_{ij}=0\text{ for all }i\neq j\text{ and,}
\displaystyle (ii)\ a_{ii}=1\text{ for all }i.

\displaystyle 9.\ \text{A matrix whose all elements are zero is called a null matrix or a}
\displaystyle \text{zero matrix.}

\displaystyle 10.\ \text{A square matrix }A=[a_{ij}]\text{ is called }
\displaystyle (i)\ \text{an upper triangular matrix, if }a_{ij}=0\text{ for all }i>j,
\displaystyle (ii)\ \text{a lower triangular matrix, if }a_{ij}=0\text{ for all }i<j.

\displaystyle 11.\ \text{Two matrices }A=[a_{ij}]_{m\times n}\text{ and }B=[b_{ij}]_{m\times n}\text{ of}
\displaystyle \text{the same order are equal, if }a_{ij}=b_{ij}\text{ for all }i=1,2,\ldots,m;
\displaystyle j=1,2,\ldots,n.

\displaystyle 12.\ \text{If }A=[a_{ij}]_{m\times n}\text{ and }B=[b_{ij}]_{m\times n}\text{ are two matrices}
\displaystyle \text{of the same order }m\times n,\text{ then their sum }A+B\text{ is an }m\times n
\displaystyle \text{matrix such that }(A+B)_{ij}=a_{ij}+b_{ij}\text{ for }i=1,2,\ldots,m\text{ and}
\displaystyle j=1,2,3,\ldots,n.\ \text{Following are the properties of matrix addition:}
\displaystyle (i)\ \text{Commutativity: If }A\text{ and }B\text{ are two matrices of the same}
\displaystyle \text{order, then }A+B=B+A.
\displaystyle (ii)\ \text{Associativity: If }A,B,C\text{ are three matrices of the same order,}
\displaystyle \text{then }(A+B)+C=A+(B+C).
\displaystyle (iii)\ \text{Existence of Identity: The null matrix is the identity element}
\displaystyle \text{for matrix addition i.e., }A+O=A=O+A.
\displaystyle (iv)\ \text{Existence of Inverse: For every matrix }A=[a_{ij}]_{m\times n}
\displaystyle \text{there exists a matrix }-A=[-a_{ij}]_{m\times n}\text{ such that}
\displaystyle A+(-A)=O=(-A)+A.
\displaystyle (v)\ \text{Cancellation Laws: If }A,B,C\text{ are three matrices of the same}
\displaystyle \text{order, then }A+B=A+C\Rightarrow B=C\text{ and }B+A=C+A\Rightarrow B=C.

\displaystyle 13.\ \text{Let }A=[a_{ij}]\text{ be an }m\times n\text{ matrix and }k\text{ be any}
\displaystyle \text{number called a scalar. Then, the matrix obtained by multiplying}
\displaystyle \text{every element of }A\text{ by }k\text{ is called the scalar multiple of }A
\displaystyle \text{by }k\text{ and is denoted by }kA.\ \text{Thus, }kA=[ka_{ij}]_{m\times n}.
\displaystyle \text{Following are the properties of scalar multiplication: If }A,B
\displaystyle \text{are two matrices of the same order and }k,l\text{ are scalars, then}
\displaystyle (i)\ k(A+B)=kA+kB\qquad (ii)\ (k+l)A=kA+lA
\displaystyle (iii)\ (kl)A=k(lA)=l(kA)\qquad (iv)\ (-k)A=-(kA)=k(-A)
\displaystyle (v)\ 1A=A\qquad (vi)\ (-1)A=-A

\displaystyle 14.\ \text{If }A\text{ and }B\text{ are two matrices of the same order, then}  A-B=A+(-B).

\displaystyle 15.\ \text{Two matrices }A\text{ and }B\text{ are conformable for the product}  AB\text{ if the number of} \\ \text{columns in }A\text{ is same as the number of} \text{rows in }B.
\displaystyle \text{If }A=[a_{ij}]_{m\times n}\text{ and }B=[b_{ij}]_{n\times p}\text{ are two} \text{matrices, then }AB\text{ is an }m\times p\text{ matrix such that}
\displaystyle (AB)_{ij}=\sum_{r=1}^{n}a_{ir}b_{rj}.
\displaystyle \text{Matrix multiplication has the following properties:}
\displaystyle (i)\ \text{Matrix multiplication is not commutative.}
\displaystyle (ii)\ \text{Matrix multiplication is associative i.e. }(AB)C=A(BC)
\displaystyle \text{wherever both sides of the equality are defined.}
\displaystyle (iii)\ \text{Matrix multiplication is distributive over matrix}
\displaystyle \text{addition i.e. }A(B+C)=AB+AC\text{ and }(B+C)A=BA+CA
\displaystyle \text{wherever both sides of the equality are defined.}
\displaystyle (iv)\ \text{If }A\text{ is an }m\times n\text{ matrix, then }I_mA=A=AI_n.
\displaystyle (v)\ \text{If }A\text{ is an }m\times n\text{ matrix and }O\text{ is a null}
\displaystyle \text{matrix, then }A_{m\times n}O_{n\times p}=O_{m\times p}\text{ and}
\displaystyle O_{p\times m}\times A_{m\times n}=O_{p\times n}.
\displaystyle \text{i.e., the product of a matrix with a null matrix is a null matrix.}

\displaystyle 16.\ \text{If }A\text{ is a square matrix, then we define }A^{1}=A \text{and }A^{n+1}=A^{n}A.

\displaystyle 17.\ \text{If }A\text{ is a square matrix and }a_{0},a_{1},\ldots,a_{n}\text{are constants, then } \\ a_{0}A^{n}+a_{1}A^{n-1}+a_{2}A^{n-2}  +\ldots+a_{n-1}A+a_{n}I\text{ is called a matrix polynomial.}

\displaystyle 18.\ \text{Let }A=[a_{ij}]\text{ be an }m\times n\text{ matrix. Then, the} \text{transpose of }A,\text{ denoted by }A^{T},\text{ is an } \\ n\times m \text{ matrix such that }(A^{T})_{ij}=a_{ji}\text{ for all }i=1,2,\ldots,m  j=1,2,\ldots,n.
\displaystyle \text{Following are the properties of transpose of a matrix:}
\displaystyle (i)\ (A^{T})^{T}=A\qquad (ii)\ (A+B)^{T}=A^{T}+B^{T}
\displaystyle (iii)\ (kA)^{T}=kA^{T}\qquad (iv)\ (AB)^{T}=B^{T}A^{T}
\displaystyle (v)\ (ABC)^{T}=C^{T}B^{T}A^{T}.

\displaystyle 19.\ \text{A square matrix }A=[a_{ij}]\text{ is called a symmetric} \text{matrix, if }a_{ij}=a_{ji}\text{ for all }i,j\text{ i.e. }A=A^{T}.

\displaystyle 20.\ \text{A square matrix }A=[a_{ij}]\text{ is called a skew}\text{symmetric matrix, if }a_{ij}=-a_{ji}\text{ for all }i,j
\displaystyle \text{i.e. }A^{T}=-A.

\displaystyle 21.\ \text{All main diagonal elements of a skew-symmetric matrix}

\displaystyle 22.\ \text{Every square matrix can be uniquely expressed as the} \text{sum of a symmetric and a} \\ \text{skew-symmetric matrix.}

\displaystyle 23.\ \text{All positive integral powers of a symmetric matrix} \text{are symmetric matrices.}

\displaystyle 24.\ \text{All odd positive integral powers of a skew-symmetric} \text{matrix are skew-symmetric matrices.}

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