1. Properties and Applications of Matrices
6.5
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2. Matrix Notation
A general element of a matrix is denoted aij. The ij refers to an element in the ith row, jth column.
For example, a31 would be an element in matrix A located in the third row, first column.
The matrices are equal if corresponding elements are equal.

3. Example: Determining matrix elements
Let aij denote a general element in A, and bij denote a general element in B, where
(a) Identify a12, b32 and a13 .
(b) Compute a31b13 + a32b23 + a33b33 .
(c) Is there a value for x that will make the statement A = B true?

4. Example: Determining matrix elements
Solution
(a) Element a12 is located in the 1st row and 2nd column of A = 3.
b32 = 2 (row 3, column 2)
a13= 4 (row 1, column 3)

5. Example: Determining matrix elements
Compute a31b13 + a32b23 + a33b33 .
= (4)(7) + (2)(–2) + (5)(2) = 34
(c) No, since a32 = 2 ≠ 5 = b32 and a33 = 5 ≠ 2 = b32 . So A ≠ B.

6. Operations on Matrices
Matrix Addition
The sum of two m  n matrices A and B is the m  n matrix A + B, in which each element is the sum of the corresponding elements of A and B. This is written as
A + B = [aij] + [bij] = [aij + bij].
If A and B have different dimensions, then A + B is undefined.

7. Operations on Matrices
Matrix Subtraction
The difference of two m  n matrices A and B is the m  n matrix A – B, in which each element is the difference of the corresponding elements of A and B. This is written as
A – B = [aij] – [bij] = [aij – bij].
If A and B have different dimensions, then A – B is undefined.

8. Operations on Matrices
Multiplication of a Matrix by a Scalar
The product of a scalar (real number) k and an m  n matrix A is the m  n matrix kA, in which each element is k times the corresponding element of A. This is written as
kA = k[aij] = [kaij].

9. Example: Performing operations on matrices
If possible, perform the indicated operations using
(a) A + 3B (b) A  C (c) –2C  3D

10. Example: Performing operations on matrices
Solution
(a) A + 3B

11. Example: Performing operations on matrices
(b) A – C is undefined because the dimension of A is 2  2 and unequal to the dimension of C, which is 3  2.

12. Example: Performing operations on matrices
(c) –2C  3D

13. Matrix Multiplication
The product of an m  n matrix A and an n  k matrix B is the m  k matrix AB, which is computed as follows. To find the element of AB in the ith row and jth column, multiply each element in the ith row of A by the corresponding element in the jth column of B. The sum of these products will give the element in row i, column j of AB.

14. Matrix Multiplication

15. Example: Multiplying matrices
If possible, compute each product using
a) AB b) CA (c) DC

16. Example: Multiplying matrices
Solution
(a) AB =

17. Example: Multiplying matrices
Solution
(b) CA =

18. Example: Multiplying matrices
Solution
c) The dimension of D is 3  3 and the dimension of C is 2  3. Therefore DC is undefined. The number of columns in D (3) does not match the number of rows in C (2).

19. Properties of Matrices
Let A, B, and C be matrices. Assume that each matrix operation is defined.
1. A + B = B + A
2. (A + B) + C = A + (B + C)
3. (AB)C = A(BC)
4. A(B + C) = AB + AC