1. MATRICES Operations with Matrices Properties of Matrix Operations
The Inverse of a Matrix
Mrs. Meena Kumari
DEPARTMENT OF ECONOMICS
PGGCG-11 ,CHANDIGARH

2. Operations with Matrices
(i, j)-th entry:
row: m
column: n
size: m×n

3. i-th row vector
row matrix
j-th column vector
Square matrix: m = n

4. Diagonal matrix:
Trace:

5. Example:

6. Equal matrix:
Example: (Equal matrix)

7. Matrix addition:
Example : (Matrix addition)

8. Scalar multiplication:
Matrix subtraction:
Ex 3: (Scalar multiplication and matrix subtraction)
Find (a) 3A, (b) –B, (c) 3A – B

9. Sol:
(a)
(b)
(c)

10. Matrix multiplication:
Size of AB
where

11. Example :
Show that AB and BA are not equal for the matrices.
and
Sol:
Note:
Note:

12. Ex : (Find AB)
Sol:

13. Properties of Matrix Operations
Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
Zero matrix:
Identity matrix of order n:

14. Properties of matrix addition and scalar multiplication:
(1) A+B = B + A
A + ( B + C ) = ( A + B ) + C
(3) 1A = A
(4) c( A+B ) = cA + cB

15. Properties of zero matrices:
Notes:
0m×n: the additive identity for the set of all m×n matrices
–A: the additive inverse of A

16. Transpose of a matrix:

17. Transpose of A matrix
(a)
(b)
(c)
Sol:
(a)
(b)
(c)
(c)

18. Properties of transposes:

19. Symmetric matrix:
A square matrix A is symmetric if A = AT
Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A
Example:
is symmetric, find a, b, c?
Sol:

20. Ex:
is a skew-symmetric, find a, b, c?
Sol:

21. The Inverse of a Matrix (1) The inverse of A is denoted by
The inverse of a matrix is unique.
(1) The inverse of A is denoted by
If A and B are invertible matrices of size n, then AB is invertible and