Math Problem Statement
Evaluate the determinant Δ given in the image.
Solution
To evaluate the determinant given in the image, we need to calculate the determinant of the 3x3 matrix:
a - b & m - n & x - y \\ b - c & n - p & y - z \\ c - a & p - m & z - x \end{vmatrix}$$ To solve this, we'll use the cofactor expansion method along the first row. The determinant of a 3x3 matrix: $$\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$$ is given by: $$\Delta = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}.$$ Now, applying this formula to our matrix: $$\Delta = (a - b) \begin{vmatrix} n - p & y - z \\ p - m & z - x \end{vmatrix} - (m - n) \begin{vmatrix} b - c & y - z \\ c - a & z - x \end{vmatrix} + (x - y) \begin{vmatrix} b - c & n - p \\ c - a & p - m \end{vmatrix}.$$ Next, let's compute the 2x2 determinants one by one: 1. **First Minor:** $$\begin{vmatrix} n - p & y - z \\ p - m & z - x \end{vmatrix} = (n - p)(z - x) - (y - z)(p - m) = nz - nx - pz + px - yp + zp + my - mz.$$ 2. **Second Minor:** $$\begin{vmatrix} b - c & y - z \\ c - a & z - x \end{vmatrix} = (b - c)(z - x) - (y - z)(c - a) = bz - bx - cz + cx - yc + za + ay - az.$$ 3. **Third Minor:** $$\begin{vmatrix} b - c & n - p \\ c - a & p - m \end{vmatrix} = (b - c)(p - m) - (n - p)(c - a) = bp - bm - cp + cm - nc + pc + an - ap.$$ Substitute these values back into the determinant expression: \[ \Delta = (a - b) [nz - nx - pz + px - yp + zp + my - mz] - (m - n) [bz - bx - cz + cx - yc + za + ay - az] + (x - y) [bp - bm - cp + cm - nc + pc + an - ap]. \] This expression represents the value of the determinant $$\Delta$$. Would you like further details on any specific part, or is there something more you would like to explore regarding this determinant? ### Related Questions: 1. What are some properties of determinants that could simplify the calculation? 2. How can we use matrix row or column operations to simplify this determinant? 3. What are some real-world applications of determinants in linear algebra? 4. How can determinants be used to find the area of a triangle or parallelogram? 5. Can we interpret this determinant geometrically in some way? **Tip:** When calculating determinants, look for opportunities to simplify the matrix using row or column operations, as these can often make the computation much easier!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Linear Algebra
Formulas
Cofactor expansion method for determinants of a 3x3 matrix
2x2 determinant formula |a b| |c d| = ad - bc
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grades 10-12