Linear Algebra & Trigonometry Study Notes

LINEAR ALGEBRA & TRIGONOMETRY Notes

SYSTEM OF EQUATIONS

• Problem Statement: Solve the following system of equations:

  • Equation 1: $x + 2y + z = 3$

  • Equation 2: $x + 3y + 7z = -1$

  • Equation 3: $3y - 2z = 7$

Method: Cramer's Rule

• Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.

• Definitions:

  • Let the system of equations be represented in matrix form as:
    $AX = B$ where:

  • A is the coefficient matrix

  • X is the column matrix of variables

  • B is the column matrix of the constants.

MATRIX OPERATIONS

• Matrix Y:

  • Matrix Y is represented as:
    $\begin{pmatrix} 15 & 9 \ 3 & 0 \ -6 & 1 \ 8 & 11 \ 9 & 10 \ -9 & 5 \ 14 & 20 \ 13 & 7 \ 6 & 3 \ 4 & 18 \ 16 & 10 \ 155 \ 921 \ \ \end{pmatrix}$

Product of Matrices

• Finding the Product: To find the product of two matrices, the inner dimensions must match; if matrix A is of size $m imes n$ and matrix B is of size $n imes p$, the resulting matrix will be of size $m imes p$.

• The product can be calculated as:
  $C_{ij} = ext{dot product of the i-th row of A and the j-th column of B}$

MATRIX INVERSION

• Problem Statement: Find the inverse of the matrix:
  $\begin{pmatrix} 4 & 18 \ 16 & 10 \ 155 \ 921 \ \ \end{pmatrix}$

Inverse Matrix Definition:

• The inverse of a matrix A, denoted as $A^{-1}$, is a matrix such that:
  $AA^{-1} = A^{-1}A = I$
  Where I is the identity matrix. Not all matrices have inverses (only non-singular matrices do). Chek the determinant to ensure it is non-zero before attempting to find the inverse.

Process to Find Inverse:

1. Calculate the determinant of the matrix.

2. If the determinant is non-zero, use the formula to find the inverse:
   $A^{-1} = rac{1}{ ext{det}(A)} ext{adj}(A)$ where adj(A) is the adjugate of A.