Math Problem Statement
solve
Solution
To solve for the characteristic determinant from the given system of linear equations, we first need to express the system as a matrix.
The given system is:
This system can be written in matrix form as:
5 & -8 \\ 10 & 5 \\ \end{pmatrix} \begin{pmatrix} I_1 \\ I_2 \\ \end{pmatrix} = \begin{pmatrix} 10 \\ 20 \\ \end{pmatrix}$$ The characteristic determinant (or simply the determinant) of the coefficient matrix is calculated as follows: $$\text{Determinant} = \text{det} \begin{pmatrix} 5 & -8 \\ 10 & 5 \\ \end{pmatrix}$$ The formula for the determinant of a 2x2 matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ is: $$\text{Determinant} = ad - bc$$ Substituting the values from our matrix: $$\text{Determinant} = (5)(5) - (10)(-8)$$ $$\text{Determinant} = 25 + 80 = 105$$ Thus, the correct answer is **D) 105**. Would you like any further details or have any questions? --- **Relative Questions:** 1. What is a determinant in the context of matrices? 2. How do you compute the determinant of a 3x3 matrix? 3. Why is the determinant important in solving systems of linear equations? 4. Can a system of equations have a determinant of zero? What does that mean? 5. How does the determinant relate to the invertibility of a matrix? 6. What are the properties of determinants under matrix multiplication? 7. How does Cramer's Rule use determinants to solve systems of linear equations? 8. How can the determinant be used to find the area of a parallelogram? **Tip:** Remember, if the determinant of a matrix is zero, the matrix is singular, meaning it does not have an inverse, and the system of equations may have no unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Determinants
Formulas
Determinant of a 2x2 matrix: det = ad - bc
Theorems
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Suitable Grade Level
High School