Problem

Find the eigenvalues and corresponding eigenvectors of the matrix A=[41 23].

Answer

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Answer

Next, we find the eigenvectors. We solve the system of linear equations (A−λI)v=0 for each eigenvalue, where v is the corresponding eigenvector. For λ1=2, we get the system [21 21][v1 v2]=[0 0]. This gives us v=[1 −2]. Similarly, for λ2=5, we get the system [−11 2−2][v1 v2]=[0 0]. This gives us v=[1 1].

Steps

Step 1 ：First we find the eigenvalues. They are the roots of the characteristic equation, det(A−λI)=0, where I is the identity matrix and λ are the eigenvalues. So, we have |4−λ1 23−λ|=0, which simplifies to (4−λ)(3−λ)−2=0. This equation simplifies to λ2−7λ+10=0, whose roots are λ1=2 and λ2=5.

Step 2 ：Next, we find the eigenvectors. We solve the system of linear equations (A−λI)v=0 for each eigenvalue, where v is the corresponding eigenvector. For λ1=2, we get the system [21 21][v1 v2]=[0 0]. This gives us v=[1 −2]. Similarly, for λ2=5, we get the system [−11 2−2][v1 v2]=[0 0]. This gives us v=[1 1].

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