Materi pembelajaran kelas XI semester 2 matematika wajib - matriks

1. ^ Rosalina Apriana – XI Science 1^
Matrix
(fun mathematics)

2. What is matrix, guys?
 A matrix is a collection of numbers arranged in rows and columns as welldenoted by uppercase. The
properties of the matrix are not the same as the table, it has rules-specific standard rules. The matrix
itself is really just the operator who becomes the machinedriving other sciences, for example
geometric transformation.
 The idea of the matrix was first introduced by Athur Cayley (1821-1895) in 1859 in England in a study of
systems of linear equations and linear transformations. However, at first, the matrix was only a game
because it could not be applied. It wasn't until 1925, 30 years after Cayley died, that matrices were
used in quantum mechanics. Furthermore, the matrix has developed rapidly and is used in various
fields.
 Matrix are usually used in table data. example: making a journal, making report cards. well, matrix
theory is usually used to add columns in the table or subtract, multiply, and divide the values in these
columns.

3. 03
Notation, order,
element,
transpose
Basic
matrix
operations
Matrix
similarity
Determinant
of the matrix
matrix constituent components
01
02 04
05
06
Invers of
matrix
Matrix
equation

4. Notation,
order, element,
transpose
01

5. 1. Notation, order and element
A matrix has many sizes. This measure is called an order. Order
innotate with Amxn where A denotes the matrix symbol, m the
number of rowsand n the number of columns.
Example : Matrix B = (2 3 0)
(-1 2 7)
The notation of the matrix above is B, while the order is B2x3,
because of the many rows2 and many columns there are 3.
2. Transpose matrix
Transpose matrix is a matrix obtained by changing the position of the
elementrows become columns and column elements become rows.
Transpose matrix Adenoted by AT. Example:
A = (1 2 4) (1 3 6)
(3 5 2) maka A^T = (2 5 7)
(6 7 3) (4 2 3)
Notation, order, element and transpose

6. Basic matrix
operations
02

7. Basic matrix operations
Multiplication
matrix
Addition and
subtraction matrices

8. Explanation matrix operations
 Multiplication scalar by matrix
Multiply the numbers distributively across the elements
Example :
C= (4 6) 3C? 3C = 3(4 6)
(7 -2) (7 -2)
 Multiplication matrix by matrix
Multiply the row elements in the left matrix by the column
elements in the right matrix, base with base, end with tip, then
sum. It must be like that, don’t do it randomly.
Example :
A = (3 5) dan B = (-4 2) find AxB?
(-2 1) (3 6)
Answer :
AxB = (3 5) ( -4 I 2) = (12+5 6+30)
-------- I (8+3 -4+6 )
(-2 1) (3 I 6)
=(3 36)
(11 2)
Addition and Substraction matrices Multiplication matrix
Add and subtract according to location
Example :
A = (4 5) dan B = (1 2)
(-2 3) (0 -9)
Find :
a. A+B
b. B-A
Answer :
a. A+B = (4 5) +(1 2) = (5 7)
(-2 3) (0 -9) (-2 -6)
b. B-A = (1 2) - (4 5) = (-3 -3)
(0 -9) (-2 3) (-2 -6)

9. Matrix
similarity
03

10. • Two matrix we can similar if all the elements have the
same value and have a symbol ‘=‘
• Example :
1. ( 3 2𝑥) (3 6)
(−2 4 ) = (-2 4), tentukan nilai x
Answer :
1. ( 3 2𝑥) ( 3 6)
2. (−2 4 ) = ( −2 4 ) Kedua ruas adalah matriks yang
sama, karena ada tanda “=”. Sehingga tinggal
disamakan saja bagian yang ingin di cari, yaitu 2𝑥 = 6
sehingga 𝑥 = 3
Matrix similarity

11. Determinant of
matrix
04

12. Determinant of matrix
● Determinan of matrix is a value that gotten by some operations of matrix.
● Determinant itself will be used to find invers.
1. Determinant of matrix by order 2x2
Determinant matrix order 2x2 if matriks 𝐴 = ( 𝑎 𝑏, 𝑐 𝑑 ) maka determinannya det 𝐴 = |𝐴| = 𝑎𝑑 − 𝑏
Example : 𝐵 = ( 2 5) maka det𝐵 = 2.6 − 4.5 = 12 − 20 = −8
( 4 6 )
2. Determinant of matrix by order 3x3
If matriks 𝐴 = ( 3 2 5 ) maka determinannya adalah :
( -4 1 2 )
( 3 3 4 )
Answer :
Sehingga det 𝐶 = 3.1.4 + 2.2.3 + 5(−4). 3 − 2(−4). 4 − 3.2.3 − 5.1.3
= 12 + 12 − 60 + 32 − 18 − 15
= −37

13. Invers of
matrix
05

14. Invers of matrix
* Invers of matrix A = (a b) is A^-1 = 1/det A. Adj A
(c d)
* Adj A = Adjoin A = (d -b)
(-c a)
*example :
find invers matrix D = (4 9)
(4 8)
Answer :
from question, det D = 32 – 36 = -4 and the adjoin = Adj D = ( 8 −9 )
(−4 4 )
So 𝐷^−1 = 1 det 𝐷 . 𝐴𝑑𝑗 𝐷 = 1 −4 ( 8 −9 −4 4 ) = ( −2 9/4)
(1 −1 )

15. Matrix
Equation
06

16. Matrix equation
• In matrix, multiplication between AxB and BxA is not same, and the value also
different, so that in matrix there are 2 formula, it is :
1. If Ax =B, so the formula is X =A^-1 . B
2. If XA = B, so the formula is X = BA^-1
• How to evidance this formula?
Remember the inverse property, if A.A-1 = I as well as A-1.A = I. Where I = identity matrix which has the
same properties as number 1. So, if we are want to solve AX = B where X is the matrix that we want to
find, then we have to eliminates A in the left side. Why can't A just be transferred to it? That because
there is no term of division between the matrix and the matrix. So that technique right to get X is to
eliminate A on the left, of course, by the rule correct, that is, by multiplying both sides by A-1 from the
left. Complete as following:
AX = B A-1.
AX = A-1.B (both sides times A-1 from the left, because A is to the left of X), the result:
I.X = A-1.B (because the nature of the identity matrix is ​​equal to 1, then I.X = X), so:
X = A-1.B

17. • Likewise for the formula for finding X at XA = B, we will see the target we want to eliminate namely A,
where is it. Right? Then we have to multiply both equations by A-1 from the right too. Produce:
XA = B XA .
A-1 = B.A-1 (both sides times A-1 from the right, because A is to the right of X), the result:
X.I = BA-1 (because the nature of the identity matrix is ​​equal to 1, then I.X = X), so:
X = B.A-1
* example :
(3 2) . X = (8 -1)
(6 -1) (x -22)
We rearrange the pattern, AX = B (A is on the left, then both sides are multiplied by A-1 from the left
Resulting in X = A-1 .B
X = A-1 .B
X = (1/det A. Adj A). B
X = 1/-15 (-1 -2) (8 -1)
(-6 3) (11 -22)
X = (2 -2)
(-1 4)

18. “If you wanna feel something extraordinary
You need to be someone incredibly.”
—rosalina apriana

19. Thanks ^^