Let A, B, C be m ×n matrices and p and q be two non-zero scalars (numbers). Then we have the following properties.

**Properties of Matrix Addition and Scalar
Multiplication**

Let *A, B, C* be *m* ×*n* matrices and *p* and
*q* be two non-zero scalars (numbers). Then we have the following
properties.

(i) *A *+* B *=* B *+*
A *[Commutative property of matrix addition]

(ii) *A *+* *(*B *+*
C*)* *=* *(*A *+* B*)* *+*C *[Associative
property of matrix addition]

(iii) ( *pq*)*A* = *p*(*qA*)
[Associative property of scalar multiplication]

(iv) *IA*=*A *[Scalar Identity
property where *I* is the unit matrix]

(v) *p*(*A *+* B*)*
*=* pA *+* pB *[Distributive property of scalar and two matrices]

(vi) ( *p* + *q* )*A* = *pA* +*qA
*[Distributive property of two scalars with a matrix]

The null matrix or zero matrix is the identity for matrix addition.

Let *A* be any matrix.

Then, *A* + *O* = *O* + *A* = *A* where *O*
is the null matrix or zero matrix of same order as that of *A*

If *A* be any given matrix then –*A* is the additive inverse of *A*.

In fact we have *A* + (−*A*) = (−*A*) + *A* = *O*

Find the value of *a, b, c, d, x, y* from the following
matrix equation.

First, we add the two matrices on both left, right hand sides to
get

Equating the corresponding elements of the two matrices, we have

*d *+* *3* *=* *2 gives *d *= –1

8 + *a* = 2*a* + 1 gives *a *= 7

3*b* − 2 = *b* – 5 gives *b *= -3/4

Substituting *a* = 7 in *a* − 4 = 4*c *gives *c*
= 3/4

Therefore, *a* = 7, *b* = − 3/2, *c* = 3/4, *d*
= –1.

**Example 3.63**

If

compute the following : (i) 3A + 2B – C (ii) 1/2 A -3/2 B

*Solution*

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10th Mathematics : Algebra : Properties of Matrix Addition and Scalar Multiplication |

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