01 Determinantlar va ularning asosiy xossalari

Summary

The lecture introduces determinants as a tool in applied mathematics, explains how to calculate small determinants, and presents properties that simplify calculations of larger ones.

Why applied mathematics matters

The lecturer explains that analyzing and modeling economic processes often requires mathematical methods. Differential calculus, limits, derivatives, and graphs can be used to study changes, extrema, rates of change, and ranges. A future specialist should understand these methods and how they are used in computer programs.

Matrices and determinants

A matrix is an ordered rectangular arrangement of numbers, similar to a table. Its size is described by its number of rows and columns. A square matrix has the same number of rows and columns, and a determinant can be calculated only for a square matrix.

For a (2\times2) matrix,

[ \begin{vmatrix} a_{11} & a_{12}\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}. ]

That is, multiply the entries on the main diagonal, then subtract the product of the entries on the other diagonal.

Calculating a (3\times3) determinant

The lecture demonstrates the triangle method, also known as Sarrus’s rule:

1. Add the products along the three diagonals running in one direction.
2. Subtract the products along the three diagonals running in the opposite direction.
3. Alternatively, repeat the first two columns beside the determinant to make the diagonal products easier to see.

This method applies to (3\times3) determinants; it is not a general method for larger determinants.

Cofactor expansion for larger determinants

For determinants of order four or greater, the lecturer introduces expansion along a chosen row or column:

1. Choose a row or column.
2. For each entry, form its minor by deleting that entry’s row and column.
3. Multiply the minor by the sign ((-1)^{i+j}), where (i) and (j) are the entry’s row and column numbers. The result is its cofactor.
4. Multiply each entry in the chosen row or column by its corresponding cofactor, then add the results.

The expansion gives the same determinant whichever row or column is selected. Choosing one with many zeros can make the calculation easier.

Main determinant properties

The lecture presents the following properties as ways to simplify calculations:

• Transpose: Switching rows and columns does not change the determinant.
• Swapping rows or columns: Interchanging two rows or two columns reverses the determinant’s sign.
• Repeated or proportional rows or columns: If two rows or two columns are identical, the determinant is zero. The same is true if one is a scalar multiple of another.
• Common factor: A factor shared by every entry in a row or column can be taken outside the determinant, scaling the determinant by that factor.
• Zero row or column: If every entry in any row or column is zero, the determinant is zero.
• Linearity in a row or column: If a row or column is a sum of two vectors, the determinant can be split into the sum of two determinants, leaving the other rows or columns unchanged.
• Adding a multiple of one row or column to another: This leaves the determinant unchanged. It is useful for creating zeros and simplifying a determinant.
• Cofactor identities: Expanding along a row or column gives the determinant. Combining the cofactors from one row or column with the entries of a different parallel row or column gives zero.

Triangular form

A determinant is triangular when all entries either above or below the main diagonal are zero. Its value is the product of the main-diagonal entries.

The lecture recommends using determinant-preserving row or column operations to turn a more complicated determinant into triangular form, then multiplying the diagonal entries. A fifth-order example illustrates this process.

Historical references and closing

The lecturer briefly discusses the development of determinant theory, naming Cardano, Leibniz, Cramer, Vandermonde, Laplace, Cauchy, Jacobi, Gauss, and the Japanese mathematician Seki Kōwa. Some dates and historical details are unclear in the subtitles.

The lecture ends by directing students to further reading and to the instructor’s email, Telegram, and institutional website.

Speakers and sources featured

• Speaker: One lecturer presents the applied mathematics lesson. The channel is identified as Akmal Abdullayev.
• Historical figures mentioned: Cardano, Gottfried Wilhelm Leibniz, Gabriel Cramer, Alexandre-Théophile Vandermonde, Pierre-Simon Laplace, Augustin-Louis Cauchy, Carl Gustav Jacob Jacobi, Carl Friedrich Gauss, and Seki Kōwa.
• Other sources: Recommended literature and the instructor’s contact and institutional web information are referenced, but the subtitles do not provide their specific titles or addresses.