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Show that the following two matrices are inverses of each other.

A = [ 1 4 −1 −3 ] , B = [ −3 −4 1 1 ]
A B = [ 1 4 −1 −3 ] [ −3 −4 1 1 ] = [ 1 ( −3 ) + 4 ( 1 ) 1 ( −4 ) + 4 ( 1 ) −1 ( −3 ) + −3 ( 1 ) −1 ( −4 ) + −3 ( 1 ) ] = [ 1 0 0 1 ] B A = [ −3 −4 1 1 ] [ 1 4 −1 −3 ] = [ −3 ( 1 ) + −4 ( −1 ) −3 ( 4 ) + −4 ( −3 ) 1 ( 1 ) + 1 ( −1 ) 1 ( 4 ) + 1 ( −3 ) ] = [ 1 0 0 1 ]
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Finding the multiplicative inverse using matrix multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication .

Finding the multiplicative inverse using matrix multiplication

Use matrix multiplication to find the inverse of the given matrix.

A = [ 1 −2 2 −3 ]

For this method, we multiply A by a matrix containing unknown constants and set it equal to the identity.

[ 1 −2 2 −3 ]     [ a b c d ] = [ 1 0 0 1 ]

Find the product of the two matrices on the left side of the equal sign.

[ 1 −2 2 −3 ]     [ a b c d ] = [ 1 a −2 c 1 b −2 d 2 a −3 c 2 b −3 d ]

Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.

1 a −2 c = 1      R 1 2 a −3 c = 0      R 2

Using row operations, multiply and add as follows: ( −2 ) R 1 + R 2 R 2 . Add the equations, and solve for c .

1 a 2 c = 1 0 + 1 c = 2 c = 2

Back-substitute to solve for a .

a −2 ( −2 ) = 1 a + 4 = 1 a = −3

Write another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.

1 b −2 d = 0 R 1 2 b −3 d = 1 R 2

Using row operations, multiply and add as follows: ( −2 ) R 1 + R 2 = R 2 . Add the two equations and solve for d .

1 b −2 d = 0 0 + 1 d = 1 d = 1

Once more, back-substitute and solve for b .

b −2 ( 1 ) = 0 b −2 = 0 b = 2
A −1 = [ −3 2 −2 1 ]
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Finding the multiplicative inverse by augmenting with the identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix A is transformed into I , the augmented matrix I transforms into A −1 .

For example, given

A = [ 2 1 5 3 ]

augment A with the identity

[ 2 1 5 3   |   1 0 0 1 ]

Perform row operations    with the goal of turning A into the identity.

  1. Switch row 1 and row 2.
    [ 5 3 2 1   |   0 1 1 0 ]
  2. Multiply row 2 by −2 and add to row 1.
    [ 1 1 2 1   |   −2 1 1 0 ]
  3. Multiply row 1 by −2 and add to row 2.
    [ 1 1 0 −1   |   −2 1 5 −2 ]
  4. Add row 2 to row 1.
    [ 1 0 0 −1   |   3 −1 5 −2 ]
  5. Multiply row 2 by −1.
    [ 1 0 0 1   |   3 −1 −5 2 ]

The matrix we have found is A −1 .

A −1 = [ 3 −1 −5 2 ]

Finding the multiplicative inverse of 2×2 matrices using a formula

When we need to find the multiplicative inverse of a 2 × 2 matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.

If A is a 2 × 2 matrix, such as

A = [ a b c d ]

the multiplicative inverse of A is given by the formula

A −1 = 1 a d b c [ d b c a ]

where a d b c 0. If a d b c = 0 , then A has no inverse.

Using the formula to find the multiplicative inverse of matrix A

Use the formula to find the multiplicative inverse of

A = [ 1 −2 2 −3 ]

Using the formula, we have

A −1 = 1 ( 1 ) ( −3 ) ( −2 ) ( 2 ) [ −3 2 −2 1 ] = 1 −3 + 4 [ −3 2 −2 1 ] = [ −3 2 −2 1 ]
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Questions & Answers

can you not take the square root of a negative number
Sharon Reply
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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