Matrix Multiplication and Solving Systems of Equations

Matrix Equations

  • A system of equations can be represented as a single matrix equation. For example:

    5x−2y=75x - 2y = 7
    3x+9y=123x + 9y = 12

    Can be written as:

    [5−2 39][x y]=[7 12]\begin{bmatrix} 5 & -2 \ 3 & 9 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 7 \ 12 \end{bmatrix}

  • This matrix equation is equivalent to the original system due to the definition of matrix multiplication.

  • If we represent matrices with capital letters, the equation becomes AX=BAX = B, where AA is the coefficient matrix, XX is the variable matrix, and BB is the constant matrix.

Solving Matrix Equations

  • To solve for the unknown matrix XX, we can use algebra:

    AX=BAX = B

    Multiply both sides by A−1A^{-1} (the inverse of AA):

    A−1AX=A−1BA^{-1}AX = A^{-1}B

    Since A−1A=IA^{-1}A = I (the identity matrix):

    IX=A−1BIX = A^{-1}B

    Therefore:

    X=A−1BX = A^{-1}B

  • If we can find A−1A^{-1}, then we can compute XX by multiplying A−1A^{-1} by BB.

Calculator Usage

  • Define the matrices AA and BB in a calculator.
  • Compute A−1BA^{-1}B to find the values of the unknown variables.

Example: 2x2 System

A=[5−2 39]A = \begin{bmatrix} 5 & -2 \ 3 & 9 \end{bmatrix}
B=[7 12]B = \begin{bmatrix} 7 \ 12 \end{bmatrix}

  • Enter matrix AA into the calculator using a command like A := [[5, -2], [3, 9]].
  • Enter matrix BB using a command like B := [[7], [12]].
  • Calculate A−1BA^{-1}B using A^(-1) * B.
  • The result will be a 2x1 matrix containing the values of xx and yy.

Example: 4x4 System

  • Consider a system of four equations with four unknowns:

    3x+2y−1z+t=73x + 2y - 1z + t = 7
    4x+1y−3z+4t=94x + 1y - 3z + 4t = 9
    2x+y+z−t=02x + y + z - t = 0
    5x+3y−z+4t=105x + 3y - z + 4t = 10

  • This can be written as a matrix equation with a 4x4 matrix.

  • Define the coefficient matrix AA:

    A=[32−11 41−34 211−1 53−14]A = \begin{bmatrix} 3 & 2 & -1 & 1 \ 4 & 1 & -3 & 4 \ 2 & 1 & 1 & -1 \ 5 & 3 & -1 & 4 \end{bmatrix}

  • Define the constant matrix BB:

    B=[7 9 0 10]B = \begin{bmatrix} 7 \ 9 \ 0 \ 10 \end{bmatrix}

  • Calculate X=A−1BX = A^{-1}B using the calculator.

  • The resulting 4x1 matrix will contain the values of xx, yy, zz, and tt.

Conclusion

  • Using matrix multiplication simplifies solving systems of equations, especially for larger systems.
  • The definition of matrix multiplication, though seemingly strange, allows us to represent systems of equations in a compact and solvable form.