91Ó°ÊÓ

Which of the points \(P(6,2,3), Q(-5,-1,4),\) and \(R(0,3,8)\) is closest to the \(x z\) -plane? Which point lies in the \(y z\) -plane?

Suppose that \(D\) is an \(n \times n\) diagonal matrix with entries \(d_{i i}\) Show that \(D^{-1}\) is an \(n \times n\) diagonal matrix with entries 1\(/ d_{i i \cdot}\)

Show that \(A=P D P^{-1},\) where \(P\) is a matrix whose columns are the eigenvectors of \(A,\) and \(D\) is a diagonal matrix with the corresponding eigenvalues. \(A=\left[ \begin{array}{ll}{1} & {2} \\ {2} & {1}\end{array}\right]\)

Suppose \(A\) is an nonsingular \(2 \times 2\) matrix. Derive the formula for its inverse, namely,$$A^{-1}=\frac{1}{a_{11} a_{22}-a_{12} a_{21}} \left[ \begin{array}{rr}{a_{22}} & {-a_{12}} \\ {-a_{21}} & {a_{11}}\end{array}\right] $$

What are the projections of the point \((2,3,5)\) on the \(x y-y z-\) and \(x z-\) planes? Draw a rectangular box with the origin and \((2,3,5)\) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Show that \(A=P D P^{-1},\) where \(P\) is a matrix whose columns are the eigenvectors of \(A,\) and \(D\) is a diagonal matrix with the corresponding eigenvalues. \(A=\left[ \begin{array}{rr}{1} & {-1} \\ {1} & {1}\end{array}\right]\)

Find all \(2 \times 2\) matrices \(A\) such that det \(A=1\) and \(A=A^{-1}\)

Show that \(A=P D P^{-1},\) where \(P\) is a matrix whose columns are the eigenvectors of \(A,\) and \(D\) is a diagonal matrix with the corresponding eigenvalues. \(A=\left[ \begin{array}{rr}{1} & {2} \\ {-3} & {3}\end{array}\right]\)

Find the transpose of each matrix. (a) \(\left[ \begin{array}{c}{3 X} \\ {1} \\ {2}\end{array}\right]\) (b) \(\left[ \begin{array}{ccc}{3} & {3} & {9}\end{array}\right]\) (c) \(\left[ \begin{array}{lll}{2} & {1} & {7} \\ {8} & {3} & {6}\end{array}\right]\) (d) \(\left[ \begin{array}{ll}{2} & {1} \\ {1} & {3}\end{array}\right]\)

Describe and sketch the surface in \(\mathbb{R}^{3}\) represented by the equation \(x+y=2 .\)