Lecture Notes: Determinants of Matrices

Introduction to Determinants

• Detailing how to find the determinant of any square matrix.
• A square matrix has an equal number of rows and columns.

Determinant of a 2x2 Matrix

• For a matrix ( \begin{bmatrix} A & B \ C & D \end{bmatrix} ), the determinant is calculated as:
  • ( \text{det} = AD - BC )
• Example:
  • Matrix: ( \begin{bmatrix} 2 & 1 \ -6 & 4 \end{bmatrix} )
  • Calculation: ( 2 \times 4 - 1 \times (-6) = 8 + 6 = 14 )
• Process:
  • Multiply the upper left and lower right entries.
  • Subtract the product of the upper right and lower left entries.
• Simple and straightforward for a 2x2 matrix.

Determinant of a 3x3 Matrix

• Matrix with entries: ( A_1, A_2, A_3, B_1, B_2, B_3, C_1, C_2, C_3 )
• The process involves breaking it down into second-order determinants (2x2 matrices).
• Steps:
  1. Take ( A_1 ) and multiply it by the determinant of the matrix formed by omitting the row and column of ( A_1 ).
  2. Subtract ( A_2 ) times the determinant of the matrix ignoring its row and column.
  3. Add ( A_3 ) times the determinant of the matrix ignoring its row and column.
• Sign alternation: Minus for the second and plus for the third entry.

Example

• Matrix: ( \begin{bmatrix} 1 & 2 & -1 \ 3 & 0 & 1 \ -5 & 4 & 2 \end{bmatrix} )
  1. First entry ( 1 ): Calculate ( 0 \times 2 - 1 \times 4 = -4 )
  2. Second entry ( 2 ): Calculate ( 3 \times 2 - 1 \times (-5) = 11 )
  3. Third entry ( -1 ): Calculate ( 3 \times 4 - 0 \times (-5) = 12 )
  4. Combine: ( 1 \times (-4) - 2 \times 11 + (-1) \times 12 = -4 - 22 - 12 = -38 )

Determinant of Larger Matrices

• Algorithm scales with the size of the matrix but follows a similar pattern.
• For a 4x4 matrix:
  • Compute using the first row, alternate signs, and compute determinants of resulting 3x3 matrices.
• Each 3x3 matrix involves determinants of three 2x2 matrices.

Conclusion

• Understanding and calculating determinants is fundamental as matrix size increases.
• Importance of organization and arithmetic accuracy.

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