What is Cramer's rule: Definition and 30 Discussions

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729).Cramer's rule implemented in a naïve way is computationally inefficient for systems of more than two or three equations. In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be numerically unstable even for 2×2 systems. However, it has recently been shown that Cramer's rule can be implemented in O(n3) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination (consistently requiring 2.5 times as many arithmetic operations for all matrix sizes), while exhibiting comparable numeric stability in most cases.

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  1. PainterGuy

    Cramer's rule and first degree polynomial curve fitting

    Hi, I did the first degree curve fitting in MATLAB. Please see below which also shows the output for each code line. But I wasn't able to get the same answer using Cramer's rule method presented below. I'm sure MATLAB answer is correct so where am I going wrong with the Cramer's rule method...
  2. Mina Farag

    Applying the implicit function theorem to a system of equations

    My attempt: According to the implicit function theorem as long as the determinant of the jacobian given by ∂(F,G)/∂(y,z) is not equal to 0, the parametrization is possible. ∂(F,G)/∂(y,z)=4yzMeaning that all points where z and y are not equal to 0 are possible parametrizations. My friend's...
  3. EEristavi

    Solving a System of Equations via the Matrix Method

    I have equation system: x + y + z - a*k = 0 -b*x + y + z = 0 -c*y + z = 0 -d*x + y = 0 where: a, b, c, d = const. Have to find: x, y, z, k Attempt of solution: I create Matrix A with coefficients; Matrix B - Solutions (Zeros) and Matrix X - variables. When I try to use Cramer's rule -...
  4. Odious Suspect

    I Cramer's Rule and Dyadics(Menzel)

    The following is from Donald H. Menzel's Mathematical Physics: ##\Phi =\left( \begin{array}{c} a_{11}\hat{\mathfrak{i}} \hat{\mathfrak{j}} +a_{12}\hat{\mathfrak{i}} \hat{\mathfrak{j}} +a_{13}\hat{\mathfrak{i}} \hat{\mathfrak{k}} \\ +a_{21}\hat{\mathfrak{j}} \hat{\mathfrak{i}}...
  5. V

    Mesh current equation in matrix form

    i3 -? This is how I tried to solve using cramer's rule (denominator is the determinant of the matrix, while numerator is the determinant of the modified matrix) I ignored the prefix Kilo, but the method is right. So I get a different value for i3. Does anyone know what's wrong?
  6. AdityaDev

    System of homogeneous equations

    I got three equations: l-cm-bn=0 -cl+m-an=0 -bl-am+n=0 In my textbook, its written "eliminating l, m, n we get:" $$ \begin{vmatrix} 1& -c& -b\\ -c& 1& -a\\ -b& -a& 1\\ \end{vmatrix}=0 $$ but if I take l, m, n as variables and since ##l=\frac{\Delta_1}{\Delta}## (Cramer's rule) and...
  7. JasonHathaway

    Differential equations and Cramer's rule

    Hi everyone, I'm taking the Differential Equations for the first time, and I want to know the most helpful textbook for the subject. We had the following example: Find the differential equation which its general solution is: y=C_{1}+C_{2} x+x^{2} Solution: y^{'}=0+C x+2x y^{''}=0+0+2...
  8. M

    Cramer's Rule and Geometry

    hey pf! so my question is how cramer's rule makes sense from a geometric perspective. I'm reading the following article: http://www.maa.org/sites/default/files/268994245608.pdf and i am good with the logic of the entire article except one point: when they say $$x=\frac{ON}{OQ}$$ can someone...
  9. F

    Using Cramer's Rule to find needed equations

    Homework Statement I am finding the minimum coefficient μ between tires and road so that cars do not slide off. I have my equations for ƩFy and ƩFx and I know μ = [fs]/[/N] So I would need to use cramer's rule on my equations to find fs and N. But I am not sure how to use cramer's rule...
  10. H

    Cramer's Rule application in differential equations

    We have this system of equations: \begin{cases} x'= -x + 2y & (1)\\ y' = -2x - y + e^{-t} & (2) \end{cases} where x(0) = 0 ; y(0) = 0 We apply the Laplace transform on (1) and (2) and get: (s + 1)X - 2Y = 0\\ 2X + (s + 1)Y = \frac{1}{s + 1} We can use elimination here...
  11. D

    Cramer's Rule and Determinants

    Homework Statement Use Cramer's rule to solve the linear system.Homework Equations (only showing one, I think if one is explained I will figure out the rest) 2x - y = -2 x + 2y = 14 What I'm told I'm supposed to do, is to take the constants accompanying the variables and make a matrix out of...
  12. Z

    Linear Equations - Cramer's Rule

    Homework Statement Does the following set of linear equations have a unique solution? (excluding any trivial solutions when x=y=z=0) Do not attempt to formally solve the equations. x+2y-4z=8 4x-6y+12z=19 -6x+3y-6z=-20 Homework Equations The Attempt at a Solution...
  13. T

    What is the Simplified Form of Cramer's Rule for Solving for y?

    Homework Statement Use Cramer's rule to solve for y (only). Call the 3x3 determinant D: (a) ax+by = 1 cx+dy=0 (b) ax+by+cz=1 dx+ey+fz=0 gx+hy+iz=0 Homework Equations Cramer's Rule: x=A-1*b The Attempt at a Solution (a) So far I have, y= [det [a 1; c...
  14. L

    Solving Linear Systems with Cramer's Rule

    Homework Statement Solve the system: ax1+3x2+x3= y1 ax2= y2 -5x1+ax2-x3=y3 using cramer's rule for those a such that the det of A is not 0 Homework Equations x1= det(A1)/det(A) det(A1)=a11+c11+a21c21+a31c31 and A1= y1 3 1 y2 a 0 y3 a 1 The Attempt at a...
  15. P

    No Solution to 3 Linear Equations Using Cramer's Rule

    while solving 3 linear equations in 3 variables by cramer's rule if all the determinant's are 0 then what can we conclude?
  16. E

    Solve using Determinants and Cramer's Rule?

    Homework Statement I was absent and missed the lecture on Cramer's rule and Determinants and have no idea how to start the homework.. The directions and problem are as follows: Using Cramer's Rule, set this problem up to find "a". Only evaluate the Denominator. When finished with the...
  17. J

    Why is Cramer's rule for determinants not 'symmetric'?

    we can solve non-homogeneous equations in matrix form using Cramer's rule. This rule is valid only if we are replacing the columns. Why can't we replace the rows and carry on the same? For eg we can use elementary transformations for obtaining inverses either via rows or via columns. But we...
  18. F

    Solving Systems of Equations: Cramer's Rule vs Standard Method

    I'm taking a circuits class, and my instructor suggests using Cramer's rule to solve systems of equations. I've just been using the standard method of Ax=b, A^{-1}b=x, where invert the A matrix and multiply it by the b vector. It seems more straight forward and is of course much more...
  19. G

    Help using Cramer's rule to show properties of determinants

    New here, have an assignment concerning Cramer's rule which I think I have a decent understanding of - I can use it to find determinants - but am a little lost on a few questions. Given the set of linear equations: a11 x1 + a12 x2 + a13 x3 = 0 a22 x2 + a23 x3 = 0...
  20. F

    Is it possable for Cramer's rule to fail?

    In a linear system of equations where there is a solution, is it possible for Cramer's rule to fail? By fail, as in end up with a zero determinate on either the top or the bottom or both.
  21. C

    Cramer's Rule Original Problem

    Homework Statement Let the curve A + By + Cx + Dy^2 + Exy + x^2 = 0 be given. It passes through the points (x_1, y_1),...,(x_5, y_5). Determine the A, B, C, D, and E. Homework Equations The Attempt at a Solution To create the system, I plug in each x_n and y_n into the...
  22. P

    Cramer's Rule in Complex Frequency Domain.

    Does Cramer's rule work in frequency domain? For example, if we have to solve two simeltaneous equations, as in nodel analysis or mesh-current method, can we use Cramer's rule to solve em? Would this be an efficient method?
  23. rocomath

    Determinants and Cramer's Rule

    I'm trying to learn about Determinants and Cramer's Rule. If a multiple of one row is added to another row, the value of the determinant is not changed. This applies to columns, also. 15 14 16 18 17 32 21 20 42 Factoring a 3 from C1 and a 2 from C3 = 6 times 5 14 13 6 17...
  24. R

    Cramer's Rule: A Simplified Explanation

    Is there any simpler proof for Cramer's rule because I've been trying to understand it from this site and some of it I just don't get http://planetmath.org/encyclopedia/ProofOfCramersRule.html
  25. N

    Using Cramer's rule to solve linear equations with complex coefficients?

    Howdy, I got down Cramer's rule down fine, now I need to extend it to include equations that have complex coefficients. Do I let each matrix entry be something like, "5 + 2i" or is there something more than that? For example, say we have (2+3i)x + (5+3i)y + (9-6i)z = 10 + i (4+3i)x + (5-3i)y...
  26. Reshma

    Solve for A using Cramer's rule with given k and K | QM problem help

    This is from a QM problem. A & B are the unknowns, k and K are given and i = \sqrt{-1}. Use Cramer's rule to find A and show that |A|2 = 1. A - B = -1 ikA - KB = ik I applied Cramer's rule to determine A: A = \frac{\left |\begin{array}{cc} -1 & -1 \\ ik & -K \end{array}\right|}{\left...
  27. H

    Limitations of Cramer's Rule in Solving Systems of Equations

    i usually solve 3 unknown sim equations using cramer's rule because it's systemic and fast, however today i encountered an equation that cramer's rule failed against age old tradition elimination and substitution method -2A + B = -2 -2A - B - C = -4 2A - 3B - C = 0 usually i don't...
  28. M

    Using cramer's Rule to solve for x2, i'm confused on this example

    I scanned the page out of the book, alittle bit is kinda burly but I'm confused on how they got A_2(B), where we are thinking of the system in matrix form AX = B. One verfies that detA = -127 and det[A_2(B)] = 313, so x2 = det[A_2(B)]/det(A) = 313/-127 = -313/127; How did they get B? and...
  29. ?

    MATLAB Using MATLAB to implement Cramer's Rule

    Hi all, I'm trying to write an m-file which will compute a solution matrix given a coefficient matrix and right hand side vector using cramer's rule.. which is no problem for a specified size matrix.. but is it possible to write an m-file which computes the solution when ANY square coefficient...
  30. Q

    Need help on matrices using cramer's rule

    I am learning how to solve a matrice using cramer's rule, and not sure if this is the correct answer. Solve the following systems of equations x - y + 3z = 8 3x + y - 2z = -2 2x + 4y + z = 0 so I figured out the solution is x = 1, y = -1, z = 2 but is this equation consider a...
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