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2. Consider the matrix A = [ 4 0 1; -2 1 0; -2 0 1 ]. (a) Find characteristic polynomial by computing det(A - ?I). This is pretty easy by either a cofactor expansion along the first row OR the diagonal method. Either way, you will initially have two terms that both have a factor of 1 - ? in them. Do not expand (multiply out) these terms - instead, factor the common factor of 1 - ? out of both, then combine and simplify the rest. (b) Give the characteristic equation for matrix A, which is obtained by setting the characteristic polynomial equal to zero. (Remember that you are doing this because the equation Ax = ?x will only have solutions x ? 0 if det(A - ?I) = 0.) Find the roots (eigenvalues) of the equation by factoring. (c) One of your eigenvalues should be one; let's refer to it as ?1. Find a basis for the eigenspace E1 corresponding to ? = 1 by solving the equation (A - I)x = 0. ((A - ?I) becomes (A - I) because ?1 = 1.) Conclude by giving the eigenspace E1 using correct notation; you can write it as an arbitrary scalar times the basis vector or as the span of the basis vector: E1 = { t [ ] } OR E1 = span ([ ]) (d) Describe the eigenspaces corresponding to the other two eigenvalues. Make it clear which eigenspace is associated with which eigenvalue. (e) You can check your answers by multiplying each eigenvector by the original matrix A to see if the result is the same as multiplying the eigenvector by the corresponding eigenvalue. In other words, if the eigenvector is x, check to see that Ax = ?x.

          2. Consider the matrix A = [ 4 0 1; -2 1 0; -2 0 1 ].

(a) Find characteristic polynomial by computing det(A - ?I). This is pretty easy by either a cofactor expansion along the first row OR the diagonal method. Either way, you will initially have two terms that both have a factor of 1 - ? in them. Do not expand (multiply out) these terms - instead, factor the common factor of 1 - ? out of both, then combine and simplify the rest.

(b) Give the characteristic equation for matrix A, which is obtained by setting the characteristic polynomial equal to zero. (Remember that you are doing this because the equation Ax = ?x will only have solutions x ? 0 if det(A - ?I) = 0.) Find the roots (eigenvalues) of the equation by factoring.

(c) One of your eigenvalues should be one; let's refer to it as ?1. Find a basis for the eigenspace E1 corresponding to ? = 1 by solving the equation (A - I)x = 0. ((A - ?I) becomes (A - I) because ?1 = 1.) Conclude by giving the eigenspace E1 using correct notation; you can write it as an arbitrary scalar times the basis vector or as the span of the basis vector:

E1 = { t [ ] } OR E1 = span ([ ])

(d) Describe the eigenspaces corresponding to the other two eigenvalues. Make it clear which eigenspace is associated with which eigenvalue.

(e) You can check your answers by multiplying each eigenvector by the original matrix A to see if the result is the same as multiplying the eigenvector by the corresponding eigenvalue. In other words, if the eigenvector is x, check to see that Ax = ?x.

2. Consider the matrix A = [ 4 0 1; -2 1 0; -2 0 1 ].

(a) Find characteristic polynomial by computing det(A - ?I). This is pretty easy by either a cofactor expansion along the first row OR the diagonal method. Either way, you will initially have two terms that both have a factor of 1 - ? in them. Do not expand (multiply out) these terms - instead, factor the common factor of 1 - ? out of both, then combine and simplify the rest.

(b) Give the characteristic equation for matrix A, which is obtained by setting the characteristic polynomial equal to zero. (Remember that you are doing this because the equation Ax = ?x will only have solutions x ? 0 if det(A - ?I) = 0.) Find the roots (eigenvalues) of the equation by factoring.

(c) One of your eigenvalues should be one; let's refer to it as ?1. Find a basis for the eigenspace E1 corresponding to ? = 1 by solving the equation (A - I)x = 0. ((A - ?I) becomes (A - I) because ?1 = 1.) Conclude by giving the eigenspace E1 using correct notation; you can write it as an arbitrary scalar times the basis vector or as the span of the basis vector:

E1 = t [ ] OR E1 = span ([ ])

(d) Describe the eigenspaces corresponding to the other two eigenvalues. Make it clear which eigenspace is associated with which eigenvalue.

(e) You can check your answers by multiplying each eigenvector by the original matrix A to see if the result is the same as multiplying the eigenvector by the corresponding eigenvalue. In other words, if the eigenvector is x, check to see that Ax = ?x.

Added by David N.

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