---
disqus: abhyas29
---
# Matrix

###### tags: `Mathematics`

[All Mathematics Formula by Abhyas here](/@abhyas/maths_formula)

Please see [README](#README) if this is the first time you are here.

## Matrix

Rectangular arrangement of $m \times n$ numbers having $m$ rows and $n$ columns. Matrix is denoted by: $\begin{bmatrix}\end{bmatrix}$ or $\begin{pmatrix}\end{pmatrix}$ or $\begin{Vmatrix}\end{Vmatrix}$

## General Form

$A=\begin{bmatrix} a_{ij} \end{bmatrix}_{m \times n}$

$a_{ij}$ is the general element

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

$\therefore A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & \ldots & a_{1n}\\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & \ldots & a_{2n}\\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & \ldots & a_{3n}\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ a_{m1} & a_{m2} & a_{m3} & a_{m4} & a_{m5} & \ldots & a_{mn}\\ \end{bmatrix}_{m \times n}$

## 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. $A=\begin{bmatrix}PD_{11} & - & -\\ - & PD_{22} & -\\ - & - & PD_{33}\end{bmatrix}_{3 \times 3}$

b. _Following Diagonal_

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

eg. $A=\begin{bmatrix}- & - & FD_{13}\\
- & FD_{22} & -\\
- FD_{31} & - & -\end{bmatrix}_{3 \times 3}$

6. **Diagonal Matrix**

Square matrix where except principal diagonal all other elements are $0$.

$A=\begin{bmatrix}d_1 & 0 & 0\\ 0 & d_2 & 0\\ 0 & 0 & d_3\end{bmatrix}_{3 \times 3}$

$A=diag(d_1, d_2, d_3)$

**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. $A=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 & a_{33}\end{bmatrix}_{3 \times 3}$

b. _Lower Triangular Matrix_

Square matrix where _except_ principal diagonal and elements below it where $i > j$, all other elements are 0.

eg. $A=\begin{bmatrix}a_{11} & 0 & 0\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & a_{33}\end{bmatrix}_{3 \times 3}$

8. **Scalar Matrix**

Diagonal Matrix where all diagonal elements are equal.

eg. $A=\begin{bmatrix}k & 0 & 0\\ 0 & k & 0\\ 0 & 0 & k\end{bmatrix}_{3 \times 3}$

9. **Identity Matrix**

Scalar matrix where all diagonal elements are $1$. It is denoted by $I_m$ where $m$ is the order.

eg. $I_3=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}$.

10. **Trace of Matrix**

Sum of diagonal elements of a square matrix

$Tr(A)=a_{11}+a_{22}+a_{33}+a_{44}+ \ldots + a_{mm}$

$\therefore Tr(A)=\sum_{i=1}^{n} a_{ij}$

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 $0$.

$\begin{vmatrix}A\end{vmatrix}=0$

12. **Non Singular Matrix**

Determinant of matrix is not equal to $0$

$\begin{vmatrix}A\end{vmatrix}\ne0$

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. $A = \begin{bmatrix}
1 & -2 & 3\\
2 & 3 & -1\\
0 & 2 & 4
\end{bmatrix}_{3 \times 3}$ and $B = \begin{bmatrix}
3 & 4 & -2\\
0 & 2 & 3\\
1 & 3 & 5
\end{bmatrix}_{3 \times 3}$

$\therefore AB = \begin{bmatrix}
(1\times3+-2\times0+3\times1) & \dots & \dots\\
\dots & \dots & \dots\\
\dots & \dots & \dots
\end{bmatrix}_{3 \times 3}$

$\therefore AB=\begin{bmatrix}
5 & 9 & 7\\
5 & 11 & 0\\
2 & 1 & 26
\end{bmatrix}_{3 \times 3}$

### 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 $A^{'}$

Change rows into columns or columns into rows.

eg. $A=\begin{bmatrix}1 & 2 & 3 & 4\\
a & b & c & d \\
\end{bmatrix}$

$\therefore A^T=\begin{bmatrix}1 & 2\\
2 & b\\
3 & c\\
4 & d\\
\end{bmatrix}$

### Properties of Transpose

1. $(A^T)^T=A$
2. $(A \pm B)^T=A^T \pm B^T$
3. $(ABC)^T = C^TB^TA^T$[Remember, product is not commutative. ] Reversal Law
4. $Tr(A) = Tr(A^T)$
5. If $A^T=A \implies A$ is symetric matrix
eg. $A=\begin{bmatrix}a & b\\
b & c\\
\end{bmatrix}$
and $A^T=\begin{bmatrix}a & b\\
b & c\\
\end{bmatrix}$
$\because A=A^T \therefore A$ is a **symetrix matrix**
6. If $A^T=-A \implies A$ is skew-symetric matrix
eg. $A=\begin{bmatrix}0 & -a& b\\
a & 0 & c\\
-b& -c& 0\\
\end{bmatrix}$
and $A^T=\begin{bmatrix}0 & a & -b\\
-a & 0 & -c\\
b & c & 0\\
\end{bmatrix}$$
$\because -A=A^T \therefore A$ is a **skew-symetrix matrix**
7. $(kA)^T=kA^T$
8. For square matrix $A$, if $AA^T=A^TA=I$ 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. $A=\begin{bmatrix}1& 4 & 5\\
3 & 2 & 6\\
0 & 1 & 0\\\end{bmatrix}$

Cofactor of middle elelment 2 would be,

$C_{22}=(-1)^{2+2}\begin{vmatrix}1 & 5 \\ 0 & 0\\\end{vmatrix}$

$\therefore C_{22}=\begin{vmatrix}1 & 5 \\ 0 & 0\\\end{vmatrix}$

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 $a_{ij}$

$\therefore\, adjA=\begin{bmatrix}C_{ij}\end{bmatrix}^T$

## Licensing and Links

[All Mathematics Formula by Abhays here](/Uhf7AR-EQcKrvHaPG9FSXg)

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