## Commutative Law:

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

The definition of subtracting two real numbers a and b is a – b = a + (-1)b or a + the opposite of b.

We can define subtraction of matrices similarly, therefore,when A and B are two matrices of the same dimensions, then

**A – B** = **A + (-1)B,** where -1 is a scalar element

For subtraction:

## Associative Law:

**A**+ (**B**+**C**) = (**A**+**B**) +**C**=**A**+**B**+**C****A**+ (-**A**) =**0**(where –**A**is the matrix composed of –aas elements)_{ij}

The properties of matrix addition and scalar multiplication are similar to the properties of addition and multiplication of real numbers. The addition of real numbers is such that the number 0 follows with the properties of additive identity. This means, *c *+ 0 = *c *for any real number. Similar properties hold for matrices:

Assuming that matrices **A**, **B** and **C** are conformable for the operations indicated and **c,d** etc. denote scalar quantity then the following are true:

**AI**=**IA**=**A**(scalar identity)**(cd)A = c(dA)**(associative property of scalar multiplication)**A + O = A**(additive identity)**c(A +B) = cA + cB**(distributive property)**(c + d)A = cA + dA**( distributive property)

**For example:**

**The general properties for matrix multiplication are as follows,**

**(AB)’ = B’A’**(multiplication and transposition)**A**(**BC**) = (**AB**)**C**=**ABC**– (associative law)**A**(**B**+**C**) =**AB**+**AC**– (first distributive law)- (
**A**+**B**)**C**=**AC**+**BC**– (second distributive law) **c(AB) = (cA)B = A(cB)**( associative property of scalar multiplication)

The division of matrices is not possible. However, *matrix inversion *works in some sense as a procedure similar to division.

The inverse of a matrix [A], expressed as **[A] ^{-1}**, is defined as:

NOTE: the inverse of a matrix [A] exists ONLY if where = the equivalent determinant of matrix [A]

**Recall that,**

**AB**may or may not be equal to**BA**, hence**BA**may not be conformable

- If
**AB**=**0**, neither**A**nor**B**necessarily be equal to**0**

- If
**AB**=**AC**,**B**may or may not be equal to =**C**

**Properties of transposed matrices:**

- (
**A**+**B**)^{T}=**A**^{T}+**B**^{T} - (
**AB**)^{T}=**B**^{T}**A**^{T} - (k
**A**)^{T}= k**A**^{T} - (
**A**^{T})^{T}=**A**