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9.7 Solving systems with inverses  (Page 3/8)

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Use the formula to find the inverse of matrix   A .   Verify your answer by augmenting with the identity matrix.

A = [ 1 −1 2     3 ]

A −1 = [ 3 5 1 5 − 2 5 1 5 ]

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Finding the inverse of the matrix, if it exists

Find the inverse, if it exists, of the given matrix.

A = [ 3 6 1 2 ]

We will use the method of augmenting with the identity.

[ 3 6 1 3       |       1 0 0 1 ]
  1. Switch row 1 and row 2.
    [ 1 3 3 6   ​ ​     ​ ​ |       0 1 1 0 ]
  2. Multiply row 1 by −3 and add it to row 2.
    [ 1 2 0 0       |       1 0 −3 1 ]
  3. There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.
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Finding the multiplicative inverse of 3×3 matrices

Unfortunately, we do not have a formula similar to the one for a   2 × 2   matrix to find the inverse of a   3 × 3   matrix. Instead, we will augment the original matrix with the identity matrix and use row operations    to obtain the inverse.

Given a   3 × 3   matrix

A = [ 2 3 1 3 3 1 2 4 1 ]

augment   A   with the identity matrix

A | I = [ 2 3 1 3 3 1 2 4 1    |    1 0 0 0 1 0 0 0 1 ]

To begin, we write the augmented matrix    with the identity on the right and   A   on the left. Performing elementary row operations    so that the identity matrix    appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.

Given a   3   ×   3   matrix, find the inverse

  1. Write the original matrix augmented with the identity matrix on the right.
  2. Use elementary row operations so that the identity appears on the left.
  3. What is obtained on the right is the inverse of the original matrix.
  4. Use matrix multiplication to show that   A A −1 = I   and   A −1 A = I .

Finding the inverse of a 3 × 3 matrix

Given the   3   ×   3   matrix   A ,   find the inverse.

A = [ 2 3 1 3 3 1 2 4 1 ]

Augment   A   with the identity matrix, and then begin row operations until the identity matrix replaces   A .   The matrix on the right will be the inverse of   A .  

[ 2 3 1 3 3 1 2 4 1       | 1 0 0 0 1 0     0 0 1 ] → Interchange  R 2   and  R 1 [ 3 3 1 2 3 1 2 4 1       | 0 1 0 1 0 0       0 0 1 ]
− R 2 + R 1 = R 1 → [ 1 0 0 2 3 1 2 4 1     |     −1 1 0 1 0 0 0 0 1 ]
− R 2 + R 3 = R 3 → [ 1 0 0 2 3 1 0 1 0     |     −1 1 0 1 0 0 −1 0 1 ]
R 3   ↔   R 2 → [ 1 0 0 0 1 0 2 3 1     |     −1 1 0 −1 0 1 1 0 0 ]
−2 R 1 + R 3 = R 3 → [ 1 0 0 0 1 0 0 3 1     |     −1 1 0 −1 0 1 3 −2 0 ]
−3 R 2 + R 3 = R 3 → [ 1 0 0 0 1 0 0 0 1     |     −1 1 0 −1 0 1 6 −2 −3 ]

Thus,

A −1 = B = [ −1   1   0 −1     0     1     6 −2 −3   ]
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Find the inverse of the   3 × 3   matrix.

A = [     2 −17 11 −1       11 −7       0           3 −2 ]

A −1 = [ 1 1     2 2 4 −3 3 6 −5 ]

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Solving a system of linear equations using the inverse of a matrix

Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:   X   is the matrix representing the variables of the system, and   B   is the matrix representing the constants. Using matrix multiplication , we may define a system of equations with the same number of equations as variables as

A X = B

To solve a system of linear equations using an inverse matrix , let   A   be the coefficient matrix    , let   X   be the variable matrix, and let   B   be the constant matrix. Thus, we want to solve a system   A X = B .   For example, look at the following system of equations.

a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2

From this system, the coefficient matrix is

A = [ a 1 b 1 a 2 b 2 ]

The variable matrix is

X = [ x y ]

And the constant matrix is

B = [ c 1 c 2 ]

Then   A X = B   looks like

[ a 1 b 1 a 2 b 2 ]     [ x y ] = [ c 1 c 2 ]

Recall the discussion earlier in this section regarding multiplying a real number by its inverse,   ( 2 −1 )   2 = ( 1 2 )   2 = 1.   To solve a single linear equation   a x = b   for   x ,   we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of   a .   Thus,

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
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Haryormhidey Reply
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Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
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Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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