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There is a way to find an inverse of a 3 ⇥ 3 matrix – or for that matter, an n ⇥ n matrix – whose determinant is not 0, but it isn't quite as simple as.
Typology: Schemes and Mind Maps
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The determinant of the 2 ⇥ 2 matrix ✓ a b c d
is the number ad cb. The above sentence is abbreviated as
det
a b c d
= ad cb
Example.
det
The determinant of a 3 ⇥ 3 matrix can be found using the formula
det
a b c d e f g h i
A (^) = a det
e f h i
b det
d f g i
d e g h
Example.
det
A (^) = 2 det
( 1) det
A matrix has an inverse exactly when its determinant is not equal to 0.
Suppose that the determinant of the 2 ⇥ 2 matrix ✓ a b c d
does not equal 0. Then the matrix has an inverse, and it can be found using the formula
✓ a b c d
det
a b c d
d b c a
Notice that in the above formula we are allowed to divide by the determi- nant since we are assuming that it’s not 0.
Example. To find ✓ 3 5 1 2
first check that
det
Then (^) ✓ 3 5 1 2
For #1-6, compute the determinant of the given matrix.
1.) (^) ✓ 2 1 1 1
(^107) p 1 0 2 2 6
7.) Which of the six matrices from the previous problems have inverses?
Find the inverses of the matrices below.
Match the functions with their graphs.
11.) f (x) = x 2 13.) p(x) =
x if x 2 ( 1, 0); x 2 if x 2 [0, 1 ).
12.) g(x) = x 14.) q(x) =
x 2 if x 2 ( 1, 0); x if x 2 [0, 1 ).
290
I
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4—’
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4—’
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(^290 27 )
I (^) r
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4—’
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