1. MATRICES
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2. After watching the slides show, you will be able to
Determine the solution of matrix problem by using
Multiple matrix operation
and inverse of matrix
and its characteristics.
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3. Multiply matrix with matrix
Look at this illustration:
Randy and Lya want to buy
book and pencil. Randy buys
3 books and 1 pencil. Lya buys
4 books and 2 pencils.
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4. How much money each of them has to pay?
If the price of a book is
Rp500,00 and
a pencil is Rp150,00;
How much money each of them has to pay?
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5. Solution: Randy = 3 x 500 + 1 x 150 = Rp1.650,00
Lya = 4 x x 150
= Rp2.300,00
The problem can be solved by using this matrix multiply:
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6. 3 1
500 4 2
150
(2 x 2)
(2 x 1)
column = row
3 x x 150 =
4 x x 150
1650 =
2300
(2 x 1)
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7. Condition of Matrix Multiply
Matrix A can be multiplied
with matrix B if
Number of matrix A column =
Number of matrix B row
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8. and matrix B has ordo n x p
If matrix A has ordo m x n
and matrix B has ordo n x p
then A x B = C
with C has ordo m x p
Am x n x Bn x p = Cm x p
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9. How to Multiply Matrix if A x B = C then matrix element of C
is the sum of multiple result of
Element of row matrix A
with element of column matrix B
which is accordance
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10. Am x n x Bn x p = Cm x p x = …………… 1st row Baris 2 … … …L U M N
2nd C O L U M N
……………
1st row x
Baris 2
… … …
1st row x 1st column
1st row x2nd column
1st row x…. =
2nd row x 1st column
2nd row x2nd column
…………..
……….x 1st column
……………..
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11. Example 1: 5 6 7 8 x
1 x x 6
1 x x 8 =
3 x x 6
3 x x 8
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12. 1 x x 6
1 x x 8 =
3 x x 6
3 x x 8 17 23 = 39 53
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13. Example 2: 1 3 2 4 x
5 x x 3
5 x x 4 =
6 x x 3
6 x x 4 26 38 = 30 44
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14. Example 3: A = and B = Evaluate: A x B and B x A Given that:
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15. 3 2 4 -1 -2 5 1 8
A x B =
3 x (-2) + (-1) x 1
3 x 5 + (-1) x 8 =
2 x (-2) + 4 x 1
2 x x 8 -7 7 42 =
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16. B x A = 3 2 4 -1 -2 5 1 8
(-2) x x 2
(-2) x (-1) + 5 x 4 =
1 x x 2
1 x (-1) + 8 x 4 4 22 = 19 31
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17. Conclusion: A x B  B x A It means matrix multiply is not commutative
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18. Example 4: The value of a from the matrices equation: is…. + =
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19. Discuss: + = = = -1 d 4 -5 2 -1 2c 1 -b 3 -3 b -4 3 c a +1 4c + (-c)
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20. 3 = 3c  c = 1 -b – 3 = -5c -b – 3 = -5 -b = -2  b = 2
3 + b = a
3 + 2 = a
5 = a
6 = 3a
So, the value of a is 2
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21. Matrices Inverse Definition: If result of multiply two matrices
is identity matrix,
(A x B = B x A = I)
then
Matrix of A is matrix inverse of B
or reverse
matrix of B is matrix inverse of A
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22. Example 1 I A = and B = A x B = = = = -5+6 -3+3 10-10 6-5
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23. Example 2 I A = and B = B x A = = = = -5+6 -15+15 2-2 6-5
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24. B = inverse of A, or A = inverse of B.
because A x B = B x A = I
it means
B = inverse of A, or A = inverse of B.
If B = inverse of A and written as A-1
then
A. A-1 = A-1. A = I
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25. Inverse Matrices (2 x 2) If A = then inverse of matrix A is A-1 =
ad – bc = determinant matrix A d -b -c a
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26. The matrix that does not have any inverse is called singular matrix
If ad – bc = 0 it means
That matrix does not have any inverse.
The matrix that does not have any inverse is called
singular matrix
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27. Example:
Given that A =
Find the inverse of matrix A
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28. Discuss: 3 -1 -5 2
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29. 1. A.A-1 = A-1.A = I 2. 3. Inverse Matrices Characteristics:
(A. B)-1 = B-1. A-1 3.
(A-1 )-1 = A
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30. Example 1
Given that A =
and B =
then (AB)-1 is … .
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31. Discuss:
AB =
-2 + 6
0 - 2
0 - 4
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32. -4 2 -6 4
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33. Example 2
If inverse of matrix A =
then find matrix A
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34. Discuss:
A = (A-1 )-1 2 -1 -4 3
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35. 16 April 2019

36. The solution of Matrices Equation If A, B and M are
matrices which has ordo(2x2)
and A is non singular matrix
then
the solutions of matrices equation are
☻AM = B is M = A-1.B
☺MA = B is M = B.A-1
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37. Example 1 Given that A = and B = Find M if: a. AM = B b. MA = B
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38. Discuss:
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39. If AM = B
then M = A-1.B
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40. b. If MA = B
then M = B.A-1
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41. Example 2 Given that multiple matrices
The value of a + b + c + d is ….
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42. Discuss:
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43. find a = 1, b = -3, c = 4 and d = 5 means
a + b + c + d = 1 – = 7
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