Algebra

Matrices

While defining a matrix we must define their order. e.g. 3 by 2 matrix i.e. rectangular array of 3 rows and 2 columns. Represented as $A = [a_{ij}]_{m*n}$ OR $A = [a_{ij}]$

Element of matrix A: $a_{11},a_{12},\dots$

Type of Matrices

Row Matrix or Row Vector: Matrix having only one row. e.g row matrix of order 1×4
A = [ 1 2 -1 -2]

Column Matrix or Column Vector: Matrix having only one column e.g column matrix of order 3×1

$A = \begin{bmatrix}1 \\0\\3\end{bmatrix}$

Square Matrix: The number of rows is equal to the number of columns. The square matrix of order n is also called the n-rowed square matrix.

Diagonal Element: Only a square matrix has the diagonal element i.e $a_{ij}$ where i=j e.g $a_{11}, a_{22},\dots$

Principal Diagonal or Leading Diagonal of the Matrix: The line along which diagonal elements lie.

Diagonal Matrix: A square matrix where all elements are zero except those in the leading diagonal i.e $a_{ij}=0$ where $i \neq j$
Denoted by A = diag[1,5,11]

$A = \begin{bmatrix}1&0&0 \\0&5&0\\0&0&11\end{bmatrix}$

Scalar Matrix: A square matrix where all diagonal elements are the same but not 0.

Identity or Unit Matrix: A square matrix where all diagonal elements are 1 and other elements are 0. The Identity matrix of order n is denoted by $I_n$ . Following is the Identity matrix of order 3

$I_3 = \begin{bmatrix}1&0&0 \\0&1&0\\0&0&1\end{bmatrix}$

Null Matrix: The matrix whose all elements are 0. Represented as 0

Upper Triangular Matrix: All elements are below the main diagonal is 0. i.e. $a_{ij} = 0$ where i>j

$A = \begin{bmatrix}1&2&4 &5\\0&5&6&3\\0&0&7&2\\0&0&0&9\end{bmatrix}$

Lower Triangular Matrix: All elements are above the main diagonal is 0. i.e. $a_{ij} = 0$ where j>i

$A = \begin{bmatrix}1&0&0 &0\\9&7&0&0\\6&2&3&0\\1&2&4 &5\end{bmatrix}$

Strict Triangular Matrix: In the Upper and Lower triangular matrix even the element in the leading diagonal is also 0

$\begin{bmatrix}0&2&4 &5\\0&0&6&3\\0&0&0&2\\0&0&0&0\end{bmatrix}\begin{bmatrix}0&0&0 &0\\9&0&0&0\\6&2&0&0\\1&2&4 &0\end{bmatrix}$

Equality of Matrices

A=[ $a_{ij}$ ] $_{m \times n}$
B=[ $b_{ij}$ ] $_{r \times s}$

If A = B then m=r and n=s i.e. their order has to be the same. And the corresponding elements are equal.

Addition of Matrices

Sum of two matrices is possible when both are of the same order
The resulting matrix also will be of the same order

Property of Matrix Addition

Theorem-1: Commutative
A+B = B+A, A& B are of the same order

Theorem-2: Associative
A, B & C are of the same order
(A+B)+C = A+(B+C)

Theorem-3: Existence of Identity
NULL matrix is the identity element for matrix addition.
A+O = A = O+A

Theorem-4: Existence of Inverse
For every matrix, there exists a matrix -A, such that
A + (-A) = O = (-A) + A
-A is called the Additive Inverse of A

Theorem-5: Cancellation Laws
A, B & C are of the same order
A+B = A+C then, B= C Left Cancellation
B+A=C+A then B=C Right Cancellation

Scalar Multiplication

Let k be any number, called Scalar. The matrix obtained by multiplying every element of A by k is called the Scalar multiple of A by k. It is denoted by kA.
kA = [k $a_{ij}$ ] $_{m \times n}$

Properties of Scalar Multiplication

A=[ $a_{ij}$ ] $_{m \times n}$
B=[ $b_{ij}$ ] $_{m \times n}$

Matrices are of the same order. k & l are the scalar.

k(A+B) = kA + kB
(k+l) A = kA + lA
(kl)A = l(kA) = k(lA)
(-k)A=-(kA) = k(-A)
IA = A
(-1)A = -A

Subtraction of Matrices

A – B = A + (-B)
A & B are of the same order

Multiplication of Matrices

A & B can be multiplied if the number of columns of A is equal to the number of the rows in B. i.e

The order of A is m x n
The order of B is n x p
The order of the product is m x p

$(AB)_{ij}$ = (i-th row A) (j-th column of B)
i = 1,2,…,m j = 1,2,…,p

$\begin{bmatrix}a_{i1}&a_{i2}&…&a_{in} \end{bmatrix}\begin{bmatrix}b_{1j}\\b_{2j}\\…\\b_{nj} \end{bmatrix}= a_{i1}b_{1j}+a_{i2}b_{2j}+\dots+a_{in}b_{nj}$

(1) $\begin{equation*} AB_{ij}=\sum_{r=1}^n{a_{ir}b_{rj}}\end{equation*}$

Properties of Matrix Multiplication

Theorem-1: Not Commutative

A = 3 x 2 B = 2 x 5
AB is possible but BA is not possible.

Theorem-2: Associative
(AB)C = A(BC)

Theorem-3: Distributive over matrix addition
A(B+C) =AB + AC

Theorem-4: If A = m x n order.
$I_m$ is the unit matrix of the column m
$I_n$ is the unit matrix of the row n
$I_m$ A=A=A $I_n$

Theorem-5: The product of a matrix with the NULL matrix is always a NULL matrix.
Note: The product of two matrices can be a NULL matrix when neither of them is a NULL matrix.

$\begin{bmatrix}0&2 \\0&0\end{bmatrix}\begin{bmatrix}1&0 \\0&0\end{bmatrix}= \begin{bmatrix}0&0 \\0&0\end{bmatrix}$

if AB is a NULL matrix then it is not necessary BA is also a NULL matrix.

Positive Integral Power of Square Matrix

$A^1$ = A
$A^{n+1} = A^nA, n \in N$
$A^mA^n = A^{mn}, m,n \in N$
$(A^m)^n = A^{mn}, m,n \in N$

Note: Applicable only for square matrix

Matrix Polynomial

$f(x) = a_0x^n+a_1x^{n-1}+a_2x^{n-2}+ \dots + a_n$
A is a square matrix

$f(A) = a_0A^n+a_1A^{n-1}+a_2A^{n-2}+ \dots + A_nI_n$

TBD

$\begin{bmatrix} 1 & 1 & 1\\1 & 1 & 2\\0&0&-1\end{bmatrix}\begin{bmatrix} 1 & 1 & 1\\0 & 2 & 2\\0&0&3\end{bmatrix}\begin{bmatrix} 1 & 2 & 5\\1 & 3 & -2\\2&4&10\end{bmatrix}$

1st Matrix: 3Row1 +2Row2 = Row3 i.e 3rd matrix dependent on first two matrices. Therefore, Singular

2nd Matrix: Row1 – Row2 = Row3 i.e 3rd matrix dependent on first two matrices. Therefore, Singular

3rd Matrix: No relation i.e 3rd matrix not dependent on first two matrices. Therefore, Non-Singular

4th Matrix: 2Row1 = Row3 i.e 3rd matrix dependent on first matrix. Therefore, Singular

The Determinant (|A|)

$\begin{bmatrix}a & b \\c & d\end{bmatrix}\begin{bmatrix}a & b & c\\d & e & f\\i & j & k\end{bmatrix}$

Singular Matrices – If the determinant is 0 then matrices are Singular
Non Singular Matrices – If the determinant is not 0 then matrices are Non Singular

For 2×2 matrix

|A| = a*d – b*c == 0 Singular

|A| = a*d – b*c != 0 Non Singular

For 3×3 matrix

|A| = a*(e*k – j* f) – b*(d*k – f*i) + c*(d*j – e* i) == 0 Singular

|A| = a*(e*k – j* f) – b*(d*k – f*i) + c*(d*j – e* i) != 0 Non Singular

Note: Shortcut to find determinant if all data below the diagonal elements are 0.

In this case determinant is the product of all diagonal element. e.g

$\begin{bmatrix}a & 0 & 0\\0 & e &0\\0 & 0 & k\end{bmatrix}$

|A| = a*e*k

Matrices

Type of Matrices

Equality of Matrices

Addition of Matrices

Property of Matrix Addition

Scalar Multiplication

Properties of Scalar Multiplication

Subtraction of Matrices

Multiplication of Matrices

Properties of Matrix Multiplication

Positive Integral Power of Square Matrix

Matrix Polynomial

TBD

The Determinant (|A|)

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