operations with 2x2 matrices ib sl/hl
TRANSCRIPT
Operations with matrices
By the end of this lesson you should be able to:• add and subtract matrices where possible• multiply matrices where possible• find the determinant of a 2x2 matrix• find the inverse of a 2x2 matrix
Adding and subtracting
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A = 2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥
B = 1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥
C = 1 2 30 4 −1
⎡
⎣ ⎢
⎤
⎦ ⎥
D =321
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
1. A+B
2. 3A-B
3. B-A
4. A+C
5. C+D
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2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥+
1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥=
3 −17 9
⎡
⎣ ⎢
⎤
⎦ ⎥
€
6 312 9
⎡
⎣ ⎢
⎤
⎦ ⎥−
1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥=
5 53 3
⎡
⎣ ⎢
⎤
⎦ ⎥
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1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥−
2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥=
−1 −3−1 3
⎡
⎣ ⎢
⎤
⎦ ⎥
Not possible
Not possible
Multiplying 2 matrices together
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A = 2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥
B = 1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥
C = 1 2 30 4 −1
⎡
⎣ ⎢
⎤
⎦ ⎥
D =321
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
1. AB
2. BA
3. AC
4. B2
5. CD
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2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥=
(2 ×1) + (1 ×3) (2 ×−2) + (1 ×6)(4 ×1) + (3 ×3) (4 ×−2) + (3 ×6)
⎡
⎣ ⎢
⎤
⎦ ⎥=
5 213 10
⎡
⎣ ⎢
⎤
⎦ ⎥
€
1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥=
−6 −430 21
⎡
⎣ ⎢
⎤
⎦ ⎥
€
2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥1 2 30 4 −1
⎡
⎣ ⎢
⎤
⎦ ⎥=
2 8 54 20 9
⎡
⎣ ⎢
⎤
⎦ ⎥
€
1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥=
−5 −1421 30
⎡
⎣ ⎢
⎤
⎦ ⎥
€
1 2 30 4 −1
⎡
⎣ ⎢
⎤
⎦ ⎥
321
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥= 13
7
⎡
⎣ ⎢
⎤
⎦ ⎥
The inverse and determinant of 2x2 matrices
1. 2x2 matrix is written as
A 2x2 matrix has a deteterminat ad-bc.
2. Swap a and d.
Make b and c negative.
3. Multiply the matrix by the 1/determinat.
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a bc d
⎡
⎣ ⎢
⎤
⎦ ⎥
€
€
A = 2 14 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
det =6 −4 =2
€
3 −1−4 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
A−1 =12
3 −1−4 2
⎡
⎣ ⎢
⎤
⎦ ⎥=
32
−12
−2 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Explain why matrix B has no inverse.
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B = 1 −23 6
⎡
⎣ ⎢
⎤
⎦ ⎥
Find the determinant, which = 0.
You can not divide the numbers by 0.
This matrix is called a singular matrix as it has no inverse.
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A = 2 34 −1
⎡
⎣ ⎢
⎤
⎦ ⎥, B = 2 5
0 −4
⎡
⎣ ⎢
⎤
⎦ ⎥, C = 1 −4
2 −8
⎡
⎣ ⎢
⎤
⎦ ⎥, D = 2 6 0
−2 −1 3
⎡
⎣ ⎢
⎤
⎦ ⎥
Now try some questions….
1. A+B
2. AB
3. A-1
4. AD
5. B+D
6. BB-1
7. C-1
8. B-1
€
4 84 −5
⎡
⎣ ⎢
⎤
⎦ ⎥
€
4 −28 24
⎡
⎣ ⎢
⎤
⎦ ⎥
€
114
314
414
− 214
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
−2 9 910 25 −3
⎡
⎣ ⎢
⎤
⎦ ⎥
Not possible
€
1 00 1
⎡
⎣ ⎢
⎤
⎦ ⎥
Not possible, as the determinant is 0.
€
48
58
0 −28
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥