Summary of "شرح المحددات مادة الرياضة معهد فني تجاري الفرقة الاولي المحاضرة (1) الشعبة العامة 2026"

Summary of the Video: شرح المحددات مادة الرياضة معهد فني تجاري الفرقة الاولي المحاضرة (1) الشعبة العامة 2026

This video is a detailed lecture by Ahmed Amboud on the topic of determinants in Business Mathematics, aimed at first-year students of the General Division at the Technical Institute of Commerce (academic year 2026). The content is also relevant for students from Faculties of Commerce, private higher institutes, and the Workers’ University. The lecturer emphasizes a clear, step-by-step manual approach to understanding and solving determinants, supplemented by calculator verification methods.

Main Ideas and Concepts

1. Introduction to Determinants

Definition: A determinant is a square matrix (same number of rows and columns) enclosed between two vertical lines.
Degree (Order) of Determinant: Determined by the number of rows (and columns).
- 2nd degree: 2x2 matrix.
- 3rd degree: 3x3 matrix.
- 4th degree: 4x4 matrix (mentioned but not elaborated).

2. Second-Degree Determinants (2x2)

Structure: Two rows and two columns.
Rule of signs: Positive main diagonal, negative secondary diagonal.
Formula for determinant value:

[ \text{Determinant} = (a \times d) - (b \times c) ]

Step-by-step solving:
- Multiply the elements of the main diagonal.
- Multiply the elements of the secondary diagonal.
- Subtract the second product from the first.
Examples given with numerical values.
Verification using calculator:
- Use calculator matrix mode (MOD → 6 → 1 or 2).
- Enter matrix values, save, and compute determinant.

3. Third-Degree Determinants (3x3)

Structure: Three rows and three columns.
Rule of signs and expansion method introduced.
Step-by-step method to calculate determinant:
- Copy the determinant and repeat the first two columns to the right side.
- Calculate the sum of the products of the diagonals going downwards (positive terms).
- Calculate the sum of the products of the diagonals going upwards (negative terms).
- Subtract the second sum from the first to get the determinant value.
Detailed example with values and manual calculation.
Verification with calculator: (MOD → 6 → 1, enter 3x3 matrix, save, compute determinant).

4. Using Determinants to Solve Systems of Linear Equations

Given linear equations, form determinants for:
- The general determinant (Δ).
- Numerator determinants for variables (Δx, Δy, Δz).
Method:
- Write the coefficient matrix as the determinant.
- To find Δx, replace the first column of coefficients with constants from the right side of equations.
- To find Δy, replace the second column similarly.
- To find Δz, replace the third column similarly.
Use the same expansion method as for 3x3 determinants.
Calculate values manually and verify with a calculator.
Explanation of the terms “simplify x” or “numerator x” as referring to these numerator determinants.

5. Calculator Usage Tips

How to enter matrices.
How to save matrices.
How to compute determinants.
Emphasis on manual calculation for institute exams but using calculators for verification or in college.

6. Additional Notes

The lecturer encourages students to study each lecture as it is released to avoid falling behind.
All materials and guides are provided for free.
Contact information and invitation for feedback via comments.

Methodologies / Step-by-Step Instructions

For 2x2 Determinants:

Write the determinant matrix:

[ \begin{vmatrix} a & b \ c & d \end{vmatrix} ]

Calculate:

[ (a \times d) - (b \times c) ]

Verify with calculator matrix mode.

For 3x3 Determinants:

Write the determinant matrix.
Copy the first two columns to the right side of the matrix.
Calculate the sum of the products of the diagonals going down:

[ a \times e \times i + b \times f \times g + c \times d \times h ]

Calculate the sum of the products of the diagonals going up:

[ c \times e \times g + a \times f \times h + b \times d \times i ]

Subtract the second sum from the first:

[ \text{Determinant} = \text{(sum down)} - \text{(sum up)} ]

Verify with calculator matrix mode.

Solving Systems of Linear Equations Using Determinants:

Form the coefficient determinant (Δ).
Form numerator determinants (Δx, Δy, Δz) by replacing the corresponding columns with the constants from the right side of the equations.
Calculate each determinant using the 2x2 or 3x3 method.
Use Cramer’s rule if needed:

[ x = \frac{\Delta x}{\Delta}, \quad y = \frac{\Delta y}{\Delta}, \quad z = \frac{\Delta z}{\Delta} ]

Speakers / Sources

Ahmed Amboud: The sole speaker and lecturer throughout the video, providing explanations, examples, and calculator demonstrations.

This summary captures the core lessons and detailed methods taught in the video, focusing on determinants and their applications in business mathematics for technical institute students.