Cramer's Rule Calculator
Tags | |
The Cramer's Rule Calculator computes the solution and determinants for two simultaneous linear equations and three simultaneous linear equations.
The Two Equation Solution uses the following form:
- a1 • x + b1 • y = c1
- a2 • x + b2 • y = c2
The solution utilizes the determinant of the 2x2 matrix.
The Three Equations Solution uses the following form:
- a1 • x + b1 • y + c1 • z = d1
- a2 • x + b2 • y + c2 • z = d2
- a3 • x + b3 • y + c3 • z = d3
The solution utilizes the determinant of the 3x3 matrix.
The Math
Derivation (2D)1
Given a system of two simultaneous equations:
`a_1 * x + b_1 *y = c_1`
`a_2 * x + b_2 *y = c_2`
we can represent these two equation in matrix form using a coefficient matrix, as `[[a_1,b_1],[a_2,b_2]] [[x],[y]] = [[c_1],[c_2]]`, where we refer to `[[a_1,b_1],[a_2,b_2]]` as the coefficient matrix.
Using Cramer's rule we compute the determinants of the coefficient matrix: `D = |[a_1,b_1],[a_2,b_2]| = a_1*b_2 - b_1*a_2`
We also form the `D_x` and `D_y` determinants as:
`D_x = |[c_1,b_1],[c_2,b_2]|` and
`D_y = |[a_1,c_1],[a_2,c_2]|`
Continuing with Cramer's Rule, we compute the values of x and y as:
`x = D_x/D`
`y = D_y/D`
Derivation (3D)
Given a system of three simultaneous equations:
`a_1 * x + b_1 *y + c_1*z = d_1`
`a_2 * x + b_2 *y + c_2*z = d_2`
`a_3 * x + b_3 *y + c_3*z = d_3`
we can represent these three equation in matrix form using a coefficient matrix, as `[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]] [[x],[y],[z]] = [[d_1],[d_2],[d_3]]`, where we refer to `[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]` as the coefficient matrix.
Using Cramer's rule we compute the determinant of the coefficient matrix: `D = |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]| = a_1*(b_2*c_3 - c_2*b_3) + b_1 * (c_2*a_3 - a_2*c_3) + c_1 * (a_2*b_3 - b_2* a_3)`
We also form the `D_x`, `D_y`, and `D_z` determinants as:
`D_x = |[d_1,b_1,c_1],[d_2,b_2,c_2],[d_3,b_3,c_3]|`
`D_y = |[a_1,d_1,c_1],[a_2,d_2,c_2],[a_3,d_3,c_3]|`
`D_z = |[a_1,b_1,d_1],[a_2,b_2,d_2],[a_3,b_3,d_3]|`
Continuing with Cramer's Rule, we compute the values of x, y, and z as:
`x = D_x/D`
`y = D_y/D`
`z = D_z/D`
Matrix Calculators
- Determinant of 3-by-3 Matrix
- Characteristic Polynomial of a 3x3 matrix
- Inverse of a 3x3 Matrix
- Transpose of a 3x3 Matrix
- Trace of a 3x3 Matrix
- Mirror of a 3x3 Matrix
- 3x3 Matrix Characteristics (Trace, Determinant, Inverse, Characteristic Polynomial)
- Product of a 3x3 matrix and a Scalar
- Product of a 3x3 matrix and a 3x1 matrix
- Product of two 3x3 matrices
- Solving 3 Equations with 3 Unknowns
- Cramer's Rule (three equations, solved for x, y and z)
- Cramer's Rule Calculator
- Illustrative video: solving system of equations with cramer's rule
