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Magazine Problem 3
Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{lll} 2 & 1 & 4 \\ 0 & 2 & 3 \\ 0 & 0 & 4 \end{array}\right], B=\left[\begin{array}{rrr} 1 & 0 & 0 \\ -1 & 4 & 0 \\ 2 & 3 & 4 \end{array}\right]$$
Problem 3
Compute the determinants in Exercises \(1-6\) using cofactor expansion along the first row and along the first column. $$\left|\begin{array}{rrr} 1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1 \end{array}\right|$$
Problem 3
Show that \(\mathbf{v}\) is an eigenvector of A and find the corresponding eigenvalue. $$A=\left[\begin{array}{rr} -1 & 1 \\ 6 & 0 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right]$$
Problem 3
Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and \((d)\) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{rrr} 1 & 1 & 0 \\ 0 & -2 & 1 \\ 0 & 0 & 3 \end{array}\right]$$
Problem 4
Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{rrr} 1 & 2 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{array}\right], B=\left[\begin{array}{lll} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right]$$
Problem 4
Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and \((d)\) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right]$$
Problem 4
Compute the determinants in Exercises \(1-6\) using cofactor expansion along the first row and along the first column. $$\left|\begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array}\right|$$
Problem 4
Show that \(\mathbf{v}\) is an eigenvector of A and find the corresponding eigenvalue. $$A=\left[\begin{array}{ll} 4 & -2 \\ 5 & -7 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 4 \\ 2 \end{array}\right]$$
Problem 5
Compute the determinants in Exercises \(1-6\) using cofactor expansion along the first row and along the first column. $$\left|\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{array}\right|$$
Problem 5
Show that \(\mathbf{v}\) is an eigenvector of A and find the corresponding eigenvalue. $$A=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 1 & -2 \\ 1 & 0 & 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right]$$
