<< Chapter < Page Chapter >> Page >
   a x = b    ( 1 a ) a x = ( 1 a ) b ( a −1     ) a x = ( a −1 ) b [ ( a −1 ) a ] x = ( a −1 ) b              1 x = ( a −1 ) b                x = ( a −1 ) b

The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.

We will investigate this idea in detail, but it is helpful to begin with a 2 × 2 system and then move on to a 3 × 3 system.

Solving a system of equations using the inverse of a matrix

Given a system of equations, write the coefficient matrix A , the variable matrix X , and the constant matrix B . Then

A X = B

Multiply both sides by the inverse of A to obtain the solution.

( A −1 ) A X = ( A −1 ) B [ ( A −1 ) A ] X = ( A −1 ) B I X = ( A −1 ) B X = ( A −1 ) B

If the coefficient matrix does not have an inverse, does that mean the system has no solution?

No, if the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.

Solving a 2 × 2 system using the inverse of a matrix

Solve the given system of equations using the inverse of a matrix.

3 x + 8 y = 5 4 x + 11 y = 7

Write the system in terms of a coefficient matrix, a variable matrix, and a constant matrix.

A = [ 3 8 4 11 ] , X = [ x y ] , B = [ 5 7 ]

Then

[ 3 8 4 11 ]     [ x y ] = [ 5 7 ]

First, we need to calculate A −1 . Using the formula to calculate the inverse of a 2 by 2 matrix, we have:

A −1 = 1 a d b c [ d b c a ]        = 1 3 ( 11 ) −8 ( 4 ) [ 11 −8 −4 3 ]        = 1 1 [ 11 −8 −4 3 ]

So,

A −1 = [ 11 −8 −4 3 ]

Now we are ready to solve. Multiply both sides of the equation by A −1 .

( A −1 ) A X = ( A −1 ) B [ 11 −8 −4 3 ]     [ 3 8 4 11 ]     [ x y ] = [ 11 −8 −4 3 ]     [ 5 7 ] [ 1 0 0 1 ]     [ x y ] = [ 11 ( 5 ) + ( −8 ) 7 −4 ( 5 ) + 3 ( 7 ) ] [ x y ] = [ −1 1 ]

The solution is ( −1 , 1 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Can we solve for X by finding the product B A −1 ?

No, recall that matrix multiplication is not commutative, so A −1 B B A −1 . Consider our steps for solving the matrix equation.

( A −1 ) A X = ( A −1 ) B [ ( A −1 ) A ] X = ( A −1 ) B I X = ( A −1 ) B X = ( A −1 ) B

Notice in the first step we multiplied both sides of the equation by A −1 , but the A −1 was to the left of A on the left side and to the left of B on the right side. Because matrix multiplication is not commutative, order matters.

Solving a 3 × 3 system using the inverse of a matrix

Solve the following system using the inverse of a matrix.

5 x + 15 y + 56 z = 35 −4 x −11 y −41 z = −26 x −3 y −11 z = −7

Write the equation A X = B .

[ 5 15 56 −4 −11 −41 −1 −3 −11 ]     [ x y z ] = [ 35 −26 −7 ]

First, we will find the inverse of A by augmenting with the identity.

[ 5 15 56 −4 −11 −41 −1 −3 −11 | 1 0 0 0 1 0 0 0 1 ]

Multiply row 1 by 1 5 .

[ 1 3 56 5 −4 −11 −41 −1 −3 −11 | 1 5 0 0 0 1 0 0 0 1 ]

Multiply row 1 by 4 and add to row 2.

[ 1 3 56 5 0 1 19 5 −1 −3 −11 | 1 5 0 0 4 5 1 0 0 0 1 ]

Add row 1 to row 3.

[ 1 3 56 5 0 1 19 5 0 0 1 5 | 1 5 0 0 4 5 1 0 1 5 0 1 ]

Multiply row 2 by −3 and add to row 1.

[ 1 0 1 5 0 1 19 5 0 0 1 5 | 11 5 −3 0 4 5 1 0 1 5 0 1 ]

Multiply row 3 by 5.

[ 1 0 1 5 0 1 19 5 0 0 1 | 11 5 −3 0 4 5 1 0 1 0 5 ]

Multiply row 3 by 1 5 and add to row 1.

[ 1 0 0 0 1 19 5 0 0 1 | −2 −3 1 4 5 1 0 1 0 5 ]

Multiply row 3 by 19 5 and add to row 2.

[ 1 0 0 0 1 0 0 0 1 | −2 −3 1 −3 1 −19 1 0 5 ]

So,

A −1 = [ −2 −3 1 −3 1 −19 1 0 5 ]

Multiply both sides of the equation by A −1 . We want A −1 A X = A −1 B :

[ −2 −3 1 −3 1 −19 1 0 5 ]     [ 5 15 56 −4 −11 −41 −1 −3 −11 ]     [ x y z ] = [ −2 −3 1 −3 1 −19 1 0 5 ]     [ 35 −26 −7 ]

Thus,

A −1 B = [ −70 + 78 −7 −105 −26 + 133 35 + 0 −35 ] = [ 1 2 0 ]

The solution is ( 1 , 2 , 0 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

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
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
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
Practice Key Terms 2

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask