Matrix

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Matrix

Rectangular arrangement of

$m\times n$ numbers having

$m$ rows and

$n$ columns. Matrix is denoted by:

$\left[\begin{array}{}\end{array}\right]$ or

$\left(\begin{array}{}\end{array}\right)$ or

$\Vert \begin{array}{}\end{array}\Vert$

General Form

$A={\left[\begin{array}{c}{a}_{ij}\end{array}\right]}_{m\times n}$

$a_{ij}$ is the general element

$m\times n$ is the order of matrix read as m corss n or m by n.

$\therefore A={\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}& \dots & a_{2n}\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}& \dots & a_{3n}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ a_{m1}& a_{m2}& a_{m3}& a_{m4}& a_{m5}& \dots & a_{mn}\end{array}\right]}_{m\times n}$

Types of Matrices

1. Row Matrix

   Matirx having one row. Order:

   $1\times n$

2. Column Matrix

   Matrix having one column. Order:

   $m\times 1$

3. Square Matrix

   Square shaped. Order:

   $m\times m$

4. Rectangular Matrix

   Rectangular shaped. Order

   $m\times n,m\ne n$

5. Diagonals

   a. Principal Dialognal

   All those elements

   $a_{ij}$ where
   $i=j$

   eg.

   $A={\left[\begin{array}{ccc}P{D}_{11}& -& -\\ -& P{D}_{22}& -\\ -& -& P{D}_{33}\end{array}\right]}_{3\times 3}$

   b. Following Diagonal

   All those Elements

   $a_{ij}$ where
   $j=m-i+1$

   eg.

   $A={\left[\begin{array}{ccc}-& -& F{D}_{13}\\ -& F{D}_{22}& -\\ -F{D}_{31}& -& -\end{array}\right]}_{3\times 3}$

6. Diagonal Matrix

   Square matrix where except principal diagonal all other elements are

   $0$.
   $A={\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]}_{3\times 3}$
   $A=diag(d_1,d_2,d_3)$

   Property

   • $A^n=diag(d_1^n,d_2^n,d_3^n)$
   • $A^{-1}=diag(d_1^{-1},d_2^{-1},d_3^{-1})$

7. Triangular Matrix

   a. Upper Triangular Matrix

   Square matrix where except principal diagonal and elements above it where

   $i<j$, all other elements are 0.

   eg.

   $A={\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right]}_{3\times 3}$

   b. Lower Triangular Matrix

   Square matrix where except principal diagonal and elements below it where

   $i>j$, all other elements are 0.

   eg.

   $A={\left[\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& 0\\ a_{31}& a_{32}& a_{33}\end{array}\right]}_{3\times 3}$

8. Scalar Matrix

   Diagonal Matrix where all diagonal elements are equal.

   eg.

   $A={\left[\begin{array}{ccc}k& 0& 0\\ 0& k& 0\\ 0& 0& k\end{array}\right]}_{3\times 3}$

9. Identity Matrix

   Scalar matrix where all diagonal elements are

   $1$. It is denoted by
   $I_m$ where
   $m$ is the order.

   eg.

   $I_3={\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}_{3\times 3}$.

10. Trace of Matrix

    Sum of diagonal elements of a square matrix

    $Tr(A)=a_{11}+a_{22}+a_{33}+a_{44}+\dots +a_{mm}$
    $\therefore Tr(A)=\sum _{i=1}^n{a}_{ij}$

    Properties of Tract of Matrix

    1. $Tr(A)=Tr(A^T)$
    2. $Tr(kA)=kTr(A)$
    3. $Tr(AB)=Tr(BA)$
    4. $Tr(I_n)=n$

11. Singular Matrix

    Square matrix whose determinant is

    $0$.
    $\left|\begin{array}{c}A\end{array}\right|=0$

12. Non Singular Matrix

    Determinant of matrix is not equal to

    $0$
    $\left|\begin{array}{c}A\end{array}\right|\ne 0$

13. Sub Matrix

    Matrix obtained by deleting rows or columns or both in given matrix A.Similar as minor in Determinant.

14. Idempotent Matrix

    Square matrix where

    $A^2=A$

15. Invoutary Matrix

    Square matrix where

    $A^2=I$

16. Nilpotent Matrix

    Square matrix of order

    $m$ where
    $A^k=0$ where
    $k$ is minimum +ve integer and
    $0$ is a null matrix

Algebra of Matrices

Equality

Two matrices are said to be equal if their order is same and corresponing elements are equal.

Addition and Subtracion

Two matrics having same order can be added or subtracted by adding or subtracting corresponing elements.

Scalar Multiplicaiton

$kA$ if found by multiplying all the element of

$A$ by

$k$

Multiplication of two matrices

Condition

$A\times B$ is only possible if

$columns(A)=rows(B)=p$(say)

Thus multiplication is possible for

$A_{mp}$ and

$B_{pn}$

Order of product will be

$m\times n$

Method

Multiplicaiton is done by taking sum of

$row(A)\times column(B)$

eg.

$A={\left[\begin{array}{ccc}1& -2& 3\\ 2& 3& -1\\ 0& 2& 4\end{array}\right]}_{3\times 3}$ and

$B={\left[\begin{array}{ccc}3& 4& -2\\ 0& 2& 3\\ 1& 3& 5\end{array}\right]}_{3\times 3}$

$\therefore AB={\left[\begin{array}{ccc}(1\times 3+-2\times 0+3\times 1)& \dots & \dots \\ \dots & \dots & \dots \\ \dots & \dots & \dots \end{array}\right]}_{3\times 3}$

$\therefore AB={\left[\begin{array}{ccc}5& 9& 7\\ 5& 11& 0\\ 2& 1& 26\end{array}\right]}_{3\times 3}$

Properties of Multiplication

1. Generally
   $AB\ne BA$
2. $(AB)C=A(BC)$ Associative
3. $A(B+C)=AB+AC$ Distributive
4. $I_m\times A_{m\times n}=A_{m\times n}=A_{m\times n}\times I_n$
5. If A is a square matrix then
   $A^{n+1}=A^n\times A$
6. $(A\pm B{)}^2=(A\pm B)\times (A\pm B)$
   
   $=A^2\pm AB\pm BA+B^2$
7. $(A+B)(A-B)=A^2-AB+BA-B^2$
8. If
   $AB=0$ does not imply that
   $A=0$ and/or
   $B=0$

Transpose of Matrix

$A^T$ or

$A^{{}^{\prime }}$

Change rows into columns or columns into rows.

eg.

$A=\left[\begin{array}{cccc}1& 2& 3& 4\\ a& b& c& d\end{array}\right]$

$\therefore A^T=\left[\begin{array}{cc}1& 2\\ 2& b\\ 3& c\\ 4& d\end{array}\right]$

Properties of Transpose

1. $(A^T{)}^T=A$
2. $(A\pm B{)}^T=A^T\pm B^T$
3. $(ABC{)}^T=C^T{B}^T{A}^T$[Remember, product is not commutative. ] Reversal Law
4. $Tr(A)=Tr(A^T)$
5. If
   $A^T=A⟹A$ is symetric matrix
   eg.
   $A=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$
   and
   $A^T=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$
   
   $\because A=A^T\therefore A$ is a symetrix matrix
6. If
   $A^T=-A⟹A$ is skew-symetric matrix
   eg.
   $A=\left[\begin{array}{ccc}0& -a& b\\ a& 0& c\\ -b& -c& 0\end{array}\right]$
   and
   $A^T=\left[\begin{array}{ccc}0& a& -b\\ -a& 0& -c\\ b& c& 0\end{array}\right]$$
   
   $\because -A=A^T\therefore A$ is a skew-symetrix matrix
7. $(kA{)}^T=k{A}^T$
8. For square matrix
   $A$, if
   $A{A}^T=A^{T}A=I$ then A is Orthogonal Matrix

Matrix to sum of symetric and skew-symetric matrix

For given matrix

$A$ do

$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$

Cofactors of elemnets of Matrix

Minor of element multiplied by

$(-1{)}^{i+j}$

eg.

$A=\left[\begin{array}{ccc}1& 4& 5\\ 3& 2& 6\\ 0& 1& 0\end{array}\right]$

Cofactor of middle elelment 2 would be,

$C_{22}=(-1{)}^{2+2}\left|\begin{array}{cc}1& 5\\ 0& 0\end{array}\right|$

$\therefore C_{22}=\left|\begin{array}{cc}1& 5\\ 0& 0\end{array}\right|$

NOTE: minor is part of determinant. Find notes for minor.

Adjoint of a Matrix

$adjA$

$A$ is a square matrix

$C_{ij}$ are the cofactors of

$a_{ij}$

$\therefore adjA={\left[\begin{array}{c}{C}_{ij}\end{array}\right]}^T$

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