Write the system of linear equations in the form A 𝐱=𝐛 and solve this matrix equation for 𝐱

step 1 There is a set of four linear equations in four variables, X1, X2, X3, and X4. So let us look at

Write the system of linear equations in the form A 𝐱=𝐛 and...

Write the system of linear equations in the form $A \mathbf{x}=\mathbf{b}$ and solve this matrix equation for $\mathbf{x}$

$$\begin{array}{rr}
2 x_{1}-x_{2}+ & x_{4}=3 \\
3 x_{2}-x_{3}-x_{4} & =-3 \\
x_{1}+x_{3}-3 x_{4} & =-4 \\
x_{1}+x_{2}+2 x_{3} & =0
\end{array}$$

Key Concepts

Solution Existence and Uniqueness

This concept addresses the conditions under which a linear system has a solution and whether that solution is unique. A system has a unique solution if the coefficient matrix is invertible and full-rank. Understanding these conditions helps in choosing an appropriate method for solving the system and in interpreting its solutions.

Matrix Inversion

When the coefficient matrix A is square and invertible, one can solve the system A x = b by computing the inverse of A and then finding x = A?¹b. This method directly yields the unique solution, though it requires that the matrix satisfies certain conditions, such as having a nonzero determinant.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by performing elementary row operations on the augmented matrix [A|b]. These operations reduce the matrix to a simpler form, either an upper-triangular or row-echelon form, from which one can easily determine the solutions via back-substitution.

Matrix Representation of Linear Systems

This concept involves rewriting a set of linear equations in compact form using matrices and vectors. In this representation, the coefficients of the variables are arranged into a matrix (commonly denoted A), the variables themselves form a vector (x), and the constants from each equation form a vector (b). This framework allows one to handle the system algebraically using the tools of linear algebra.

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