Linear Algebra and Matrix Operations

Matrix Operations

• A matrix is in reduced row echelon form (RREF) if it meets certain conditions.
• The matrix product [\begin{pmatrix} 1 & 2 & \sqrt{6}\ 0 & 0 & 1\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 3 & 0 & 1\ 6 & -1 & 2\ 2 & 0 & 1 \end{pmatrix}] is in RREF, and √ is equal to 1 + √6.

Linear Systems

• A linear system consists of multiple equations with variables.
• The linear system x + y + z = a, 2x + 3y + 3z = 16, 5x + 2y + 3z = 11 has a solution x = -1, y = 2, and z = r, where a ∈ ℝ and r ∈ ℝ, and a is equal to -5.
• The linear system x + 2y + 5z = 2, 4x + y - 15z = 15, 2x + 3y + 5z = 5 has infinitely many solutions.
• The linear system x + y + 3z = 2, 4x + 5y + 2z = 4, 3x + y + 29z = 14 has a solution z = t, t ∈ ℝ, and y is equal to -4 - 10t, and x is equal to 13 - 6t.

Linear System Applications

• The linear system x + 2y - z + 3w = -5, 2x + 5y + 2z + 7w = -3, 3x + 8y + 6z + 9w = -2, -2x - 5y + z -12w = -1 has a solution with a value of w equal to -1, and a value of z equal to 3.