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Solving Systems of Linear Equations
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A system of linear equations consists of two or more linear equations with the same set of variables. Solving such a system means finding values for the variables that satisfy all equations simultaneously. Matrices provide powerful and systematic methods for solving these systems.
A general system of $n$ linear equations with $n$ unknowns can be written as: $$ a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1 $$ $$ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2 $$ $$ \vdots $$ $$ a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n = b_n $$ This can be represented in matrix form as $AX = B$, where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $B$ is the column vector of constants.
Solving Equations: Matrix Inverse Method
2x2 System Solver (Matrix Inverse)
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3x3 System Solver (Matrix Inverse)
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If a system of linear equations is represented as $AX = B$, and if the coefficient matrix $A$ is square and its inverse $A^{-1}$ exists (i.e., $\det(A) \neq 0$), then the solution is given by $X = A^{-1}B$.
For a 2x2 system:
Given: $$ a_{11}x + a_{12}y = b_1 $$ $$ a_{21}x + a_{22}y = b_2 $$ Matrix form: $ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} $
The inverse $A^{-1}$ is $ \frac{1}{\det(A)} \begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix} $, where $\det(A) = a_{11}a_{22} - a_{12}a_{21}$.
Solving Equations: Cramer's Rule
Cramer's Rule is another method to solve a system $AX=B$ using determinants. If $\det(A) \neq 0$, then the unique solution is given by: $$ x_i = \frac{\det(A_i)}{\det(A)} $$ where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the column vector $B$.
For a 2x2 system:
$x = \frac{\det(A_x)}{\det(A)} = \frac{\begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix}}{\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$, $y = \frac{\det(A_y)}{\det(A)} = \frac{\begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix}}{\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$
2x2 System Solver (Cramer's Rule)
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Solving Equations: Gaussian Elimination
Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It involves transforming the augmented matrix $[A|B]$ into row-echelon form using elementary row operations, and then using back-substitution to find the solution.
Elementary row operations include:
- Swapping two rows.
- Multiplying a row by a non-zero scalar.
- Adding a multiple of one row to another row.
2x2 System Solver (Gaussian Elimination)
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Eigenvalues and Eigenvectors
For a square matrix $A$, an eigenvector $v$ and its corresponding eigenvalue $\lambda$ satisfy the equation $Av = \lambda v$, where $v$ is a non-zero vector.
This can be rewritten as $(A - \lambda I)v = 0$, where $I$ is the identity matrix. For non-trivial solutions for $v$, the determinant of $(A - \lambda I)$ must be zero: $$ \det(A - \lambda I) = 0 $$ This is called the characteristic equation, and its roots are the eigenvalues $\lambda$. Once eigenvalues are found, they are substituted back into $(A - \lambda I)v = 0$ to find the corresponding eigenvectors $v$.