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Chapter 3: Problem 59
Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} 4 u & -1 \\ -1 & 2 v \end{array}\right|$$
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Let \(A_{11}, A_{12},\) and \(A_{22}\) be \(n \times n\) matrices. Find the determinant of the partitioned matrix $$\left[\begin{array}{cc}A_{11} & A_{12} \\ 0 & A_{22}\end{array}\right]$$ in terms of the determinants of \(A_{11}, A_{12},\) and \(A_{22}\)
Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right] ; \quad \lambda_{1}=1, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] \\ \lambda_{2}=2, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -7 \\ 4 \\ 1 \end{array}\right] \end{array}$$
Determine whether the points are collinear. $$(-2,5),(0,-1),(3,-9)$$
Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text (a) The characteristic equation of the matrix \(A=\left[\begin{array}{rr}2 & -1 \\\ 1 & 0\end{array}\right]\) yields eigenvalues \(\lambda_{1}=\lambda_{2}=1\) (b) The matrix \(A=\left[\begin{array}{rr}4 & -2 \\ -1 & 0\end{array}\right]\) has irrational eigenvalues \(\lambda_{1}=2+\sqrt{6}\) and \(\lambda_{2}=2-\sqrt{6}\)
Prove that if \(A\) is an orthogonal matrix, then \(|A|=\pm 1\)
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