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How is linear algebra used to solve systems of linear equations?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, systems of linear equations are typically represented using matrix-vector notation as $A X = B$ , where $A$ is the matrix of coefficients, $X$ is the column vector of unknown variables, and $B$ is the column vector of constants.12

Several methods are used to solve these systems:

Gaussian Elimination: This is a primary numerical method that involves reducing the system through a sequence of operations—solving one equation for a variable and substituting it into others—until the variables can be determined via "back substitution". It is highly efficient and often implemented using matrix operations.31
Matrix Inversion: If the coefficient matrix $A$ is square and nonsingular (meaning its determinant is non-zero), the system can be solved by finding the inverse matrix $A^{- 1}$ . The solution is then given by $X = A^{- 1} B$ .2
Cramer’s Rule: This procedure uses determinants to solve systems of equations. While useful for theoretical study and small systems, it is generally not considered an effective method for systems involving more than three variables.4

For larger or more complex systems, numerical analysts may also employ iterative methods to approximate solutions with increasing accuracy.1