A system of linear equations can be represented using matrices by utilizing the coefficient matrix, the variable matrix, and the constant matrix.

  Let's consider a system of linear equations with 'n' variables and 'm' equations:

  a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁

  a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂

  .

  .

  .

  aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ

  To represent this system using matrices, we can define the coefficient matrix (A), the variable matrix (X), and the constant matrix (B).

  The coefficient matrix (A) is an 'm x n' matrix where each element aᵢⱼ represents the coefficient of the variable xⱼ in the i-th equation.

  The variable matrix (X) is an 'n x 1' matrix where each element xⱼ represents the value of the j-th variable.

  The constant matrix (B) is an 'm x 1' matrix where each element bᵢ represents the constant term on the right-hand side of the i-th equation.

  Thus, the system of linear equations can be rewritten using matrices as:

  AX = B

  This is the matrix form of the system of linear equations. By performing matrix operations such as multiplication and inversion, we can solve for the variable matrix (X) and obtain the solution to the system of linear equations.

  It's important to note that the system of linear equations must be consistent and solvable for this matrix representation to be valid. If the coefficient matrix (A) is singular (non-invertible), there may not be a unique solution to the system.