Determinants:

• Minors:
  • For a 2x2 matrix, the minor of an element is the determinant of the submatrix formed by deleting the row and column containing that element.
  • For a 3x3 matrix, the minor of an element is the determinant of the submatrix formed by deleting the row and column containing that element, multiplied by the sign factor $(-1)^{i+j}$, where (i) is the row index and (j) is the column index of the element.
  • For a 4x4 matrix, the minor of an element is the determinant of the submatrix formed by deleting the row and column containing that element, multiplied by the sign factor $(-1)^{i+j+k}$, where (i) is the row index, (j) is the column index, and (k) is the index of the minor in the row/column expansion (e.g., 1 for the first minor in the first row expansion, 2 for the second minor in the first row expansion, and so on).
• Cofactors:
  • The cofactor of an element in a matrix is the signed minor of that element. The sign is determined by the sign factor $(-1)^{i+j}$, where (i) is the row index and (j) is the column index of the element.
• Expansion by any row/column:
  • The determinant of a matrix can be calculated by expanding along any row or column. This involves multiplying each element in the chosen row/column by its corresponding cofactor and summing the results.
• Evaluation using Laplace’s expansion:
  • Laplace’s expansion is a method for calculating the determinant of a matrix by repeatedly applying row/column expansions. It involves selecting a row/column that contains a large number of zeros or small values to minimize the computational effort.
• Determinant of a triangular matrix:
  • The determinant of a triangular matrix (lower or upper triangular) is simply the product of its diagonal elements.
• Determinant of a diagonal matrix:
  • The determinant of a diagonal matrix is the product of its diagonal elements.
• Singular and non-singular matrices:
  • A square matrix is singular if its determinant is zero, and non-singular if its determinant is non-zero.
  • A singular matrix does not have an inverse, while a non-singular matrix has an inverse.
• Product of determinants:
  • The determinant of the product of two matrices is equal to the product of their determinants.
• Inverse of a matrix using determinants (Cramer’s rule):
  • Cramer’s rule can be used to find the inverse of a square matrix by using determinants. It involves calculating the determinant of the matrix and the determinants of the matrices obtained by replacing each column of the original matrix with the column vector representing the unknown variables.
• Applications in solving systems of linear equations:
  • Determinants can be used to solve systems of linear equations by reducing them to matrix equations and using Cramer’s rule to find the solutions.
• Applications in geometry:
  • Determinants are used in geometry to find areas/volumes of parallelograms/parallelepipeds by calculating the determinant of the matrix representing the coordinates of the vertices.

Numerical Problems:

• Finding the values of determinants: Calculate the determinants of the following matrices:

$$A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix}. $$

• Solving equations using determinants: Solve the following system of linear equations using Cramer’s rule:

$$3x + 2y = 5$$

$$2x - y = 3$$

• Applying determinants in geometry: Find the area of the parallelogram with vertices (A(1, 2), B(3, 4), C(6, 5), ) and (D(4, 3)).