To compute the determinant of the lower matrix using the determinant of the upper matrix, let's first analyze the situation:

1. The upper determinant involves a block matrix with values such as $8x, -4r, -4u,$ etc., and we are likely meant to compute it symbolically or reduce it using properties.
2. The lower determinant (of the matrix $\begin{bmatrix} x & y & z \\ u & v & w \\ r & s & t \end{bmatrix}$) can be derived by substituting information from the upper determinant.

Steps to solve:

1. Determine the determinant of the upper matrix symbolically or explicitly using the given equation.
2. Use any relationships or values implied by the first determinant to calculate the second determinant.

Let me extract the key equations and calculate. One moment.The upper matrix in your image represents a determinant equation that involves values $x, y, z, r, s, t, u, v, w$. This determinant equals $-56$. I will outline the steps to proceed with the calculations.

1. Structure of the Matrices:

• The upper determinant is of the form: 8x & -4r & -4u \\ -2y + 2x & s - r & v - u \\ -2z & t & w \end{vmatrix} = -56$$
• The lower determinant to calculate is: x & y & z \\ u & v & w \\ r & s & t \end{vmatrix}.$$

2. Approach to Solve:

• Expand the upper determinant symbolically to find relationships among $x, y, z, r, s, t, u, v, w$.
• Compare terms with the lower determinant if applicable.

I'll now proceed to calculate. sympy import symbols, Matrix, det

Define symbols for variables

x, y, z, r, s, t, u, v, w = symbols('x y z r s t u v w')

Upper determinant matrix

upper_matrix = Matrix([ [8x, -4r, -4u], [-2y + 2x, s - r, v - u], [-2z, t, w] ])

Lower determinant matrix

lower_matrix = Matrix([ [x, y, z], [u, v, w], [r, s, t] ])

Calculate the determinants

upper_determinant = det(upper_matrix) lower_determinant = det(lower_matrix)

upper_determinant, lower_determinant### Results:

1. Upper Determinant: $\text{Upper Determinant} = 8(rvz - rwy - suz + swx + tuy - tvx)$ It is essentially 8 times the lower determinant.

2. Lower Determinant: $\text{Lower Determinant} = rvz - rwy - suz + swx + tuy - tvx$

Using the relationship:

The upper determinant equals $-56$. Since the upper determinant is $8 \times \text{Lower Determinant}$, we have: $8 \times \text{Lower Determinant} = -56$ Thus: $\text{Lower Determinant} = -\frac{56}{8} = -7$

Final Answer:

The determinant of the lower matrix is: $\boxed{-7}$

Would you like further clarification or details?

Related Questions:

1. What is the geometric interpretation of a determinant in 3D space?
2. How does scaling rows or columns affect the determinant?
3. How can we verify this relationship using linear transformations?
4. What happens if one row or column in the matrix becomes zero?
5. Can this method generalize to larger block matrices?

Tip:

Always check for scalar multiplication factors when comparing determinants of related matrices. It simplifies calculations significantly!