If **A, B, C are m × n matrix** and c and d are scalars, the following properties are true.

(1) commutative property with respect to addition

**A+B= B+A**

(2) associative property with respect to addition

**(A+B)+C= A+(B+C)**

(3) associative property of scalar addition

**(c d) A=c (d A)** where c and d are scalar.

(4) Existence of additive identity OR null matrix

**A+0= 0+A =A** (where 0 is null matrix)

(5) Existence of **multiplicative identity**(unit/identity or imaginary matrix)

**1A= A1= A**

(6) Distributive property w.r.t scalar multiplication.

**(a) c (A+B) = c A + c. B**

**(b) (c+d) A= c A + d A**

(7) associative property with respect to multiplication.

**A(BC) = (AB)C**

(8) Left distribution Law (property)

**A(B+C)= AB+AC**

(9) Right distribution property

**(A+B)C = AC+BC**

(10) Right scalar distribution property.

**c (AB) = (c A)B= A(c B)**

(11) diagonal matrix properties

A **diagonal matrix i**s called an identity matrix. If all diagonal entries are 1

EXAMPLE

**EQUAL MATRIX:**

any two matrices A and B are called equal matrix, if

**(i) order of A = order of B**

**(ii) when corresponding entries are same**

**ADDITIVE INVERSE MATRIX:**

Let A be a matrix. A matrix B is defined as an additive inverse of A. If

**B+A=0= A+B**

**DETERMINANT OF M:**

A real number (lambda) is called **determinant of M,** denoted by M

**ADJOINT OF M MATRIX:**

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