Matrix Form of a 3x3 Linear System

Overview of the Three-by-Three (3×33 \times 3) System

  • The transcript introduces a transition into working with a system of three linear equations with three variables (3×33 \times 3 system).
  • The primary objective of this section is to demonstrate the process of writing this linear system in matrix form.
  • The speaker notes an expectation that at this stage, the students should be proficient enough to understand and execute the writing of these systems into matrix notation without extensive remediation.

Mathematical Construction of a 3×33 \times 3 System

  • A general 3×33 \times 3 system of linear equations is typically represented as follows:
    • Equation 1: a11x+a12y+a13z=b1a_{11}x + a_{12}y + a_{13}z = b_1
    • Equation 2: a21x+a22y+a23z=b2a_{21}x + a_{22}y + a_{23}z = b_2
    • Equation 3: a31x+a32y+a33z=b3a_{31}x + a_{32}y + a_{33}z = b_3
  • Components of the system:
    • x,y,zx, y, z: These represent the unknown variables (or the unknowns vector).
    • aija_{ij}: These represent the numerical coefficients where ii denotes the row (equation number) and jj denotes the column (variable index).
    • b1,b2,b3b_1, b_2, b_3: These represent the constant terms residing on the right-hand side of the equals sign.

Representation in Matrix Form (AX=BAX = B)

  • The linear system is converted into a compact matrix equation expressed as AX=BAX = B.

  • The Coefficient Matrix (AA):

    • This is a square matrix composed of the coefficients of the variables.
    • For the 3×33 \times 3 system, it is defined as:
    • A=(a11a12a13a21a22a23a31a32a33)A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}

  • The Variable Vector (XX):

    • This is a column vector (also called a column matrix) containing the variables for which the solution is sought.
    • It is expressed as:
    • X=(xyz)X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}

  • The Constant Vector (BB):

    • This is a column vector containing the constants from the right side of the equations.
    • It is expressed as:
    • B=(b1b2b3)B = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}

Procedural Requirements for Matrix Representation

  • Alignment:
    • To correctly transcribe the system into matrix AA, variables must be aligned vertically in the same order across all equations (e.g., all xx terms in the first column, all yy terms in the second, and all zz terms in the third).
  • Zero Coefficients:
    • If a specific variable is missing from one of the equations in the system, a coefficient of 00 must be used as a placeholder in the corresponding position within matrix AA.
  • The Augmented Matrix Alternative:
    • While the transcript focuses on the matrix form AX=BAX = B, these systems are often also written as an augmented matrix for calculation purposes:
    • (A∣B)=(a11a12a13∣b1a21a22a23∣b2a31a32a33∣b3)(A|B) = \begin{pmatrix} a_{11} & a_{12} & a_{13} & | & b_1 \\ a_{21} & a_{22} & a_{23} & | & b_2 \\ a_{31} & a_{32} & a_{33} & | & b_3 \end{pmatrix}