Welcome to the **Linear Algebra Calculator**. Use this tool to quickly and accurately solve a 3×3 system of linear equations using Cramer’s Rule. Input the coefficients and constant terms below to find the unique solution for the variables $x$, $y$, and $z$.
Linear Algebra Calculator: 3×3 System Solver
Enter the coefficients ($a_{ij}$) and constant terms ($b_i$) for the system of equations:
$a_{11}x + a_{12}y + a_{13}z = b_1$
$a_{21}x + a_{22}y + a_{23}z = b_2$
$a_{31}x + a_{32}y + a_{33}z = b_3$
Linear Algebra Calculator Formula: Cramer’s Rule (3×3)
This calculator uses Cramer’s Rule, which employs determinants to solve systems of linear equations. For the system $Ax=B$, the solution for a variable $x_i$ is given by:
$$x = D_x / D \quad y = D_y / D \quad z = D_z / D$$
Where D is the determinant of the coefficient matrix A, and $D_x$, $D_y$, $D_z$ are the determinants of the matrices formed by replacing the respective coefficient column with the constant vector B.
Formula Source (1): Wolfram MathWorld – Cramer’s Rule
Formula Source (2): Paul’s Online Math Notes – Systems of Equations
Variables: Input Explanation
To solve the system, you must input 12 coefficients/constants corresponding to the following variables:
- $a_{11}, a_{12}, a_{13}, a_{21}, \dots, a_{33}$ (Coefficients): The numerical factors multiplying the variables ($x$, $y$, $z$) in each equation.
- $b_1, b_2, b_3$ (Constants): The values on the right-hand side of the equal sign for each of the three equations.
What is Linear Algebra?
Linear algebra is a branch of mathematics concerned with vector spaces, linear mappings, and systems of linear equations. It is foundational to nearly all areas of modern mathematics, physics, engineering, and data science, including machine learning and computer graphics.
Unlike basic algebra, which deals with individual numbers and simple equations, linear algebra provides the tools to handle and manipulate large collections of variables and relationships simultaneously, primarily through the use of matrices and vectors.
How to Calculate a 3×3 System of Linear Equations (Example)
Using the example system $2x + y – z = 8$, $-3x – y + 2z = -11$, and $-2x + y + 2z = -3$, here are the steps using Cramer’s Rule:
- Form the Coefficient Matrix (A): Construct the 3×3 matrix from the $x, y, z$ coefficients: $$\begin{pmatrix} 2 & 1 & -1 \\ -3 & -1 & 2 \\ -2 & 1 & 2 \end{pmatrix}$$
- Calculate the Determinant (D): Calculate the determinant of Matrix A. If $D=0$, there is no unique solution. In this case, $D = 10$.
- Calculate $D_x, D_y, D_z$: Replace the corresponding columns in A with the constant vector $B = \begin{pmatrix} 8 \\ -11 \\ -3 \end{pmatrix}$ and calculate the new determinants. $D_x = 20, D_y = 10, D_z = -10$.
- Find the Solution: Divide each determinant by the main determinant D: $$x = D_x / D = 20 / 10 = 2$$ $$y = D_y / D = 10 / 10 = 1$$ $$z = D_z / D = -10 / 10 = -1$$
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Frequently Asked Questions (FAQ)
What is a determinant in linear algebra?
The determinant is a scalar value that is a function of the entries of a square matrix. It provides vital information about the matrix, such as whether the matrix is invertible (non-singular). If the determinant is zero, the matrix is singular, and the system of equations has either no solution or infinitely many solutions.
Can this calculator solve non-square systems?
No, this calculator is specifically designed to solve square systems (3 equations and 3 variables) using Cramer’s Rule. Non-square systems require different methods like Gaussian elimination or least-squares approximation.
When should I use Gaussian Elimination instead of Cramer’s Rule?
While Cramer’s Rule is excellent for small systems (like 2×2 or 3×3), it becomes computationally very inefficient for larger matrices. For systems of size 4×4 or larger, Gaussian Elimination or LU decomposition is generally preferred due to much lower computational cost.
What does it mean if the determinant is zero?
If the main determinant (D) is zero, the coefficient matrix is singular. This means the system of equations is linearly dependent, and you cannot find a unique solution for $x, y,$ and $z$. The calculator will return an error in this case.