Simplify the Matrix M=[[1,-4],[2,-7]]

Problem

M=[[1,−4],[2,−7]]

Solution

1. Identify the given matrix M Since the task is to simplify the matrix, we look for properties such as the determinant, the inverse, or the characteristic equation to describe it in its simplest terms.

2. Calculate the determinant of the matrix M using the formula det(M)=a*d−b*c

det(M)=(1)*(−7)−(−4)*(2)

det(M)=−7+8

det(M)=1

3. Find the inverse of the matrix M Since the determinant is 1 the inverse M(−1) is found by swapping the main diagonal elements and changing the signs of the off-diagonal elements.

M(−1)=1/1*[[−7,4],[−2,1]]

M(−1)=[[−7,4],[−2,1]]

4. Determine the characteristic equation by solving det(M−λ*I)=0

det(1−λ)=0

(1−λ)*(−7−λ)−(−8)=0

λ2+6*λ+1=0

5. Solve for eigenvalues using the quadratic formula.

λ=(−6±√(,36−4))/2

λ=−3±2√(,2)

Final Answer

M=[[1,−4],[2,−7]]

Want more problems? Check here!