This document discusses the properties of matrices, including properties of matrix addition and scalar multiplication. For matrix addition, it describes the commutation law, associative law, existence of an additive identity and inverse, and cancellation laws. For scalar multiplication, it lists four properties. It also provides three examples calculating matrices and solving systems of equations involving matrices.

1. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Tutorial-IV: Matrices
Basic Mathematics:
Properties of Matrices
Dr. Mehar Chand
Department of Mathematics
Baba Farid College, Bathinda-151001
April 23, 2020
Dr. Mehar Chand Properties of Matrices

2. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Contents
1 Properties of Matrix Addition
Commutation Law
Associative Law
Existence of Additive Identity
Existence of Additive Inverse
Cancellation Laws
2 Properties of Scalar Multiplication
3 Examples
4 Full video Lecture
Dr. Mehar Chand Properties of Matrices

3. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Commutative Law)
If A and B are matrices of order m × n, then A + B = B + A
Dr. Mehar Chand Properties of Matrices

4. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Associative Law)
If A, B and C are matrices of order m × n, then
(A + B) + C = A + (B + C)
Dr. Mehar Chand Properties of Matrices

5. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Existence of Additive Identity)
For every Matrix A, there exist a matrix O such that
A + O = A = O + A
Dr. Mehar Chand Properties of Matrices

6. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Existence of Additive Inverse)
For every Matrix A, there exist a matrix B such that
A + B = O = B + A
Dr. Mehar Chand Properties of Matrices

7. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Cancellation Laws)
If A, B and C are three matrices of same order, then
A + B = A + C ⇒ B = C [ Left Cancellation Law]
B + A = C + A ⇒ B = C [ Right Cancellation Law]
Dr. Mehar Chand Properties of Matrices

8. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Properties of Matrix Addition
Definition (Properties of Scalar Multiplication)
If A and B are matrices of same order and k, l are scalars,
then
1 k(A + B) = kA + kB
2 (k + l)A = kA + lA
3 k(lA) = (kl)A
4 (−k)A = −(kA) = k(−A)
Dr. Mehar Chand Properties of Matrices

9. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Examples
Example (1)
If A =


1 2 −3
5 0 2
1 −1 1

, B =


3 −1 2
4 2 5
2 0 3

 and
C =


4 1 2
0 3 2
1 −2 3

, then compute A + B and B − C Also
verify that A + (B − C) = (A + B) − C
Dr. Mehar Chand Properties of Matrices

10. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Examples
Example (2)
If A =


8 0
4 −2
3 6

 and B =


2 −2
4 2
−5 1

, then find the
matrix X such that 2A + 3X = 5B
Example (3)
Find X and Y, If 2X + 3Y =
2 3
4 0
and
3X + 2Y =
2 −2
−1 5
.
Dr. Mehar Chand Properties of Matrices

11. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Full video Lecture
To see full video lecture visit the below link
https://youtu.be/hEId2ZuD4Qc
Dr. Mehar Chand Properties of Matrices

12. Properties of
Matrices
Dr. Mehar
Chand
Properties of
Matrix
Addition
Commutation Law
Associative Law
Existence of Identity
Existence of Inverse
Cancellation Laws
Scalar
Multiplication
Examples
Video
Thank you
Dr. Mehar Chand Properties of Matrices