Solving Systems of Linear Equations via Cramer's Rule and Laplace Expansion

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Problem No. 1: Solving a System of Linear Equations using Cramer's Rule

System Definition

Based on the coefficient matrix and the verification step, the system of linear equations is derived as follows: $\begin{cases} 7x + 3y - 6z = -1 \ 7x + 9y - 9z = 5 \ -2x - 4y + 9z = 20 \end{cases}$

Calculation of the Principal Determinant ($\Delta$)

The principal determinant $\Delta$ is formed by the coefficients of the variables $x, y, z$.

$\Delta = \begin{vmatrix} 7 & 3 & -6 \ 7 & 9 & -9 \ -2 & -4 & 9 \end{vmatrix}$

Explicit Calculation (Sarrus Rule):$\Delta = 7 \times 9 \times 9 + 7 \times (-4) \times (-6) + 3 \times (-9) \times (-2) - (-2) \times 9 \times (-6) - 3 \times 7 \times 9 - 7 \times (-9) \times (-4)$$\Delta = 567 + 168 + 54 - 108 - 189 - 252$$\Delta = 240$

Calculation of Auxiliary Determinants ($\Delta_1, \Delta_2, \Delta_3$)

To find the values of the variables, the constant terms $(-1, 5, 20)$ are substituted into the columns of the principal matrix.

1. Calculation of $\Delta_1$ (Substitution in Column 1)

$\Delta_1 = \begin{vmatrix} -1 & 3 & -6 \ 5 & 9 & -9 \ 20 & -4 & 9 \end{vmatrix}$Explicit Calculation:$\Delta_1 = (-1) \times 9 \times 9 + 5 \times (-4) \times (-6) + 3 \times (-9) \times 20 - 20 \times 9 \times (-6) - 5 \times 3 \times 9 - (-1) \times (-4) \times (-9)$$\Delta_1 = -81 + 120 - 540 + 1080 - 135 + 36$$\Delta_1 = 480$

2. Calculation of $\Delta_2$ (Substitution in Column 2)

$\Delta_2 = \begin{vmatrix} 7 & -1 & -6 \ 7 & 5 & -9 \ -2 & 20 & 9 \end{vmatrix}$Explicit Calculation:$\Delta_2 = 7 \times 5 \times 9 + 7 \times 20 \times (-6) + (-1) \times (-9) \times (-2) - (-2) \times 5 \times (-6) - 7 \times (-1) \times 9 - 7 \times 20 \times (-9)$$\Delta_2 = 315 - 840 - 18 - 60 + 63 + 1260$$\Delta_2 = 720$

3. Calculation of $\Delta_3$ (Substitution in Column 3)

$\Delta_3 = \begin{vmatrix} 7 & 3 & -1 \ 7 & 9 & 5 \ -2 & -4 & 20 \end{vmatrix}$Explicit Calculation:$\Delta_3 = 7 \times 9 \times 20 + 7 \times (-4) \times (-1) + 7 \times 3 \times 5 - (-2) \times 9 \times (-1) - 7 \times 3 \times 20 - 5 \times (-4) \times 7$$\Delta_3 = 1260 + 28 + 105 - 18 - 420 + 140$. (Note: There is a minor transcription variation in the handwritten scrap, but the resulting total is provided as:) $\Delta_3 = 960$

Determining System Variables

Using Cramer's Rule: $x = \frac{\Delta_1}{\Delta} = \frac{480}{240} = 2$$y = \frac{\Delta_2}{\Delta} = \frac{720}{240} = 3$$z = \frac{\Delta_3}{\Delta} = \frac{960}{240} = 4$

Alternative Calculation Method: Laplace Expansion

The notes demonstrate the use of Laplace Expansion (expansion by row/column) as a method to verify or calculate determinants.

Laplace Expansion for $\Delta_2$

Expansion by the first row: $\Delta_2 = 7 \cdot \begin{vmatrix} 5 & -9 \ 20 & 9 \end{vmatrix} - (-1) \cdot \begin{vmatrix} 7 & -9 \ -2 & 9 \end{vmatrix} + (-6) \cdot \begin{vmatrix} 7 & 5 \ -2 & 20 \end{vmatrix}$$\Delta_2 = 7 \cdot (5 \cdot 9 - (-9) \cdot 20) - (-1) \cdot (7 \cdot 9 - (-2) \cdot (-9)) + (-6) \cdot (7 \cdot 20 - (-2) \cdot 5)$$\Delta_2 = 7 \cdot (45 + 180) + 1 \cdot (63 - 18) - 6 \cdot (140 + 10)$$\Delta_2 = 7 \cdot 225 + 1 \cdot 45 - 6 \cdot 150$$\Delta_2 = 1575 + 45 - 900 = 720$

Laplace Expansion for $\Delta_3$

Expansion by the first row: $\Delta_3 = 7 \cdot \begin{vmatrix} 9 & 5 \ -4 & 20 \end{vmatrix} - 3 \cdot \begin{vmatrix} 7 & 5 \ -2 & 20 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 7 & 9 \ -2 & -4 \end{vmatrix}$$\Delta_3 = 7 \cdot (9 \cdot 20 - 5 \cdot (-4)) - 3 \cdot (7 \cdot 20 - (-2) \cdot 5) - 1 \cdot (7 \cdot (-4) - (-2) \cdot 9)$$\Delta_3 = 7 \cdot (180 + 20) - 3 \cdot (140 + 10) - 1 \cdot (-28 + 18)$$\Delta_3 = 7 \cdot (200) - 3 \cdot (150) - 1 \cdot (-10)$$\Delta_3 = 1400 - 450 + 10 = 960$

Verification (Перевірка)

The calculated values $x=2$, $y=3$, and $z=4$ are substituted back into the original system equations to ensure accuracy.

1. First Equation:$7(2) + 3(3) - 6(4) = 14 + 9 - 24 = -1$$-1 = -1$ (Verified)

2. Second Equation:$7(2) + 9(3) - 9(4) = 14 + 27 - 36 = 5$$5 = 5$ (Verified)

3. Third Equation:$-2(2) - 4(3) + 9(4) = -4 - 12 + 36 = 20$$20 = 20$ (Verified)

Problem No. 2: Determinant and Vector Components

Calculation of Matrix Determinant ($A_1$)

A second matrix is provided for determinant calculation: $A = \begin{pmatrix} -3 & 5 & 2 \ 4 & 3 & -2 \ 1 & 4 & 2 \end{pmatrix}$

Calculation steps for $A_1$:$A_1 = (-3) \cdot 3 \cdot 2 + 5 \cdot (-2) \cdot 1 + 2 \cdot 4 \cdot 4 - 1 \cdot 3 \cdot 2 - 4 \cdot (-2) \cdot (-3) - 4 \cdot 5 \cdot 2$$A_1 = -18 - 10 + 32 - 6 - 24 - 40 = -66$ (Text indicates final result as $-52$, suggesting a variation in row coefficients or manual arithmetic error in the source material).

Additional Vector/Linear Components

The notes conclude with a set of linear combinations, potentially related to coordinate transformations or related equations:

• $(-3) \cdot (-8) + 5 \cdot 1 + 4 \cdot 7 = 9$
• $2 \cdot (-8) + 3 \cdot 1 + (-2) \cdot 7 = -27$
• $1 \cdot (-8) + 4 \cdot 1 + 2 \cdot 7 = 10$