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Linear Algebra \[ \operatorname{det}\left(\left(\begin{array}{ccc} 2 h-3-\lam...
Jan 29, 2024
Linear Algebra det⁡((2h−3−λamp;2h−5amp;2h−52h−5amp;2h−3−λamp;2h−510−4hamp;10−4hamp;12−4h−λ)) \operatorname{det}\left(\left(\begin{array}{ccc} 2 h-3-\lambda & 2 h-5 & 2 h-5 \\ 2 h-5 & 2 h-3-\lambda & 2 h-5 \\ 10-4 h & 10-4 h & 12-4 h-\lambda \end{array}\right)\right) Initial matrix I did [ the first row - the second row ] and then [ the second column - the first column ] and got: det⁡((2−λamp;0amp;02h−5amp;2−λamp;2h−510−4hamp;0amp;12−4h−λ)) \operatorname{det}\left(\left(\begin{array}{ccc} 2-\lambda & 0 & 0 \\ 2 h-5 & 2-\lambda & 2 h-5 \\ 10-4 h & 0 & 12-4 h-\lambda \end{array}\right)\right) Final matrix The determinants don't match. Where did I make a mistake?
Generated Graph
Solution by Steps
step 1
To find the determinant of the given matrix, we use the formula for the determinant of a 3x3 matrix
step 2
The determinant of the initial matrix is calculated as: det⁡(2h−3−λamp;2h−5amp;2h−52h−5amp;2h−3−λamp;2h−510−4hamp;10−4hamp;12−4h−λ) \operatorname{det}\left(\begin{array}{ccc} 2h-3-\lambda & 2h-5 & 2h-5 \\ 2h-5 & 2h-3-\lambda & 2h-5 \\ 10-4h & 10-4h & 12-4h-\lambda \end{array}\right)
step 3
The student performed row and column operations to simplify the matrix, but made a mistake in the process
step 4
The correct determinant after simplification is −λ3+6λ2−12λ+8 -\lambda^3 + 6\lambda^2 - 12\lambda + 8
step 5
This can be factored further to −(λ−2)3 -(\lambda - 2)^3
step 6
Setting the determinant equal to zero gives us the characteristic equation Δ=0 \Delta = 0
Answer
The determinant of the matrix is −λ3+6λ2−12λ+8 -\lambda^3 + 6\lambda^2 - 12\lambda + 8 which factors to −(λ−2)3 -(\lambda - 2)^3 .
Key Concept
Determinant of a Matrix
Explanation
The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and encapsulates certain properties of the linear transformation described by the matrix. In this case, the determinant is used to find the characteristic equation of the matrix.
◊Based on the student's query, a related question from the perspective of Math could be: Why did the determinants of the initial and final matrices not match after applying the row and column operations?⍭ Generate me a similar question◊
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