1. Home
  2. College algebra
  3. Systems of equations and
  4. Solving systems with cramer's

7.8 Solving systems with cramer's rule  (Page 4/11)

<< Chapter < Page
  College algebra     Page 4 / 11
Chapter >> Page >

Understanding properties of determinants

There are many properties of determinants . Listed here are some properties that may be helpful in calculating the determinant of a matrix.

Properties of determinants

  1. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.
  2. When two rows are interchanged, the determinant changes sign.
  3. If either two rows or two columns are identical, the determinant equals zero.
  4. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.
  5. The determinant of an inverse matrix   A − 1   is the reciprocal of the determinant of the matrix   A .
  6. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.

Illustrating properties of determinants

Illustrate each of the properties of determinants.

Property 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the main diagonal.

A = [ 1     2 3 0     2 1 0     0 − 1 ]

Augment   A   with the first two columns.

A = [ 1 2 3 0 2 1 0 0 − 1 |       1 0 0         2 2 0 ]

Then

det ( A ) = 1 ( 2 ) ( −1 ) + 2 ( 1 ) ( 0 ) + 3 ( 0 ) ( 0 ) − 0 ( 2 ) ( 3 ) − 0 ( 1 ) ( 1 ) + 1 ( 0 ) ( 2 )                                 = −2

Property 2 states that interchanging rows changes the sign. Given

A = [ −1 5 4 −3 ] ,     det ( A ) = ( −1 ) ( −3 ) − ( 4 ) ( 5 ) = 3 − 20 = −17 B = [ 4 − 3 − 1 5 ] ,     det ( B ) = ( 4 ) ( 5 ) − ( −1 ) ( −3 ) = 20 − 3 = 17

Property 3 states that if two rows or two columns are identical, the determinant equals zero.

                      A = [ 1 2 2 2 2 2 −1 2 2    |    1 2 −1   2 2 2 ] det ( A ) = 1 ( 2 ) ( 2 ) + 2 ( 2 ) ( −1 ) + 2 ( 2 ) ( 2 ) + 1 ( 2 ) ( 2 ) − 2 ( 2 ) ( 1 ) − 2 ( 2 ) ( 2 )                               = 4 − 4 + 8 + 4 − 4 − 8 = 0

Property 4 states that if a row or column equals zero, the determinant equals zero. Thus,

A = [ 1 2 0 0 ] ,       det ( A ) = 1 ( 0 ) − 2 ( 0 ) = 0

Property 5 states that the determinant of an inverse matrix   A − 1   is the reciprocal of the determinant   A .   Thus,

            A = [ 1 2 3 4 ] , det ( A ) = 1 ( 4 ) − 3 ( 2 ) = −2 A − 1 = [ − 2 1 3 2 − 1 2 ] , det ( A − 1 ) = − 2 ( − 1 2 ) − ( 3 2 ) ( 1 ) = − 1 2

Property 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Thus,

A = [ 1 2 3 4 ] , det ( A ) = 1 ( 4 ) − 2 ( 3 ) = −2 B = [ 2 ( 1 ) 2 ( 2 ) 3 4 ] , det ( B ) = 2 ( 4 ) − 3 ( 4 ) = −4
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using cramer’s rule and determinant properties to solve a system

Find the solution to the given 3 × 3 system.

2 x + 4 y + 4 z = 2 ( 1 ) 3 x + 7 y + 7 z = −5 ( 2 )    x + 2 y + 2 z = 4 ( 3 )

Using Cramer’s Rule    , we have

D = | 2 4 4 3 7 7 1 2 2 |

Notice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out.

  1. Multiply equation (3) by –2 and add the result to equation (1).
    − 2 x − 4 y − 4 x = − 8     2 x + 4 y + 4 z = 2                                                                         0 = − 6

Obtaining a statement that is a contradiction means that the system has no solution.

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

Access these online resources for additional instruction and practice with Cramer’s Rule.

Key concepts

  • The determinant for   [ a b c d ]   is   a d − b c .   See [link] .
  • Cramer’s Rule replaces a variable column with the constant column. Solutions are   x = D x D , y = D y D .   See [link] .
  • To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See [link] .
  • To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution:   x = D x D , y = D y D , z = D z D .   See [link] .
  • Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See [link] and [link] .
  • Certain properties of determinants are useful for solving problems. For example:
    • If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.
    • When two rows are interchanged, the determinant changes sign.
    • If either two rows or two columns are identical, the determinant equals zero.
    • If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.
    • The determinant of an inverse matrix   A − 1   is the reciprocal of the determinant of the matrix   A .
    • If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. See [link] and [link] .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask