Math Problem Statement
Use the properties of determinants to show that the determinant of the given 4x4 matrix equals abc.
Solution
To solve this determinant problem and show that the determinant equals , let us proceed step-by-step using properties of determinants:
Matrix Provided:
1 & 1 & 1 & 1 \\ 1 & 1+a & 1 & 1 \\ 1 & 1 & 1+b & 1 \\ 1 & 1 & 1 & 1+c \end{vmatrix}$$ ### Step 1: Subtract the first row from each of the other rows Using the property of determinants that allows row operations (without changing the value of the determinant), we perform the following transformations: - Row 2 becomes $$R_2 - R_1$$, - Row 3 becomes $$R_3 - R_1$$, - Row 4 becomes $$R_4 - R_1$$. The resulting matrix is: $$\begin{vmatrix} 1 & 1 & 1 & 1 \\ 0 & a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c \end{vmatrix}$$ ### Step 2: Expand the determinant This matrix is now upper triangular (all entries below the diagonal are zero). For an upper triangular matrix, the determinant is simply the product of the diagonal elements: $$\text{Determinant} = 1 \cdot a \cdot b \cdot c = abc$$ ### Final Answer: $$\boxed{abc}$$ Would you like a deeper explanation of any step? Let me know! --- Here are five related questions to explore this topic further: 1. What are the key properties of determinants that simplify calculations like this? 2. How does subtracting rows affect the determinant of a matrix? 3. Can you compute the determinant of a $$3 \times 3$$ matrix using the same properties? 4. What happens to the determinant if you multiply one row by a scalar? 5. How would the determinant change if the diagonal elements included $$a+b+c$$ instead of separate terms? ### Tip: When simplifying determinants, try to create zeros using row or column operations to reduce computation complexity!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Linear Algebra
Matrix Row Operations
Formulas
Det(A) = product of diagonal elements for upper triangular matrices
Theorems
Properties of Determinants
Row Operation Invariance
Suitable Grade Level
College Level or Advanced High School