91Ó°ÊÓ

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrr}5 & 2 & 1 \\\\-10 & -2 & 3 \\\15 & 2 & -3\end{array}\right]$$

Verify that the given vector function \(\mathbf{x}\) defines a solution to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b}\) for the given \(A\) and \(\mathbf{b}.\) $$\begin{aligned}&\mathbf{x}(t)=\left[\begin{array}{c}e^{4 t} \\\\-2 e^{4 t}\end{array}\right], A=\left[\begin{array}{rr}2 & -1 \\\\-2 & 3\end{array}\right]\\\&\mathbf{b}(t)=\left[\begin{array}{l}0 \\\0\end{array}\right]\end{aligned}.$$

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.) \(4 \times 4\) skew-symmetric matrix.

Use the LU factorization of \(A\) to solve the system \(A \mathbf{x}=\mathbf{b}\). $$A=\left[\begin{array}{rrrr}4 & 3 & 0 & 0 \\\8 & 1 & 2 & 0 \\\0 & 5 & 3 & 6 \\\0 & 0 & -5 & 7 \end{array}\right], \mathbf{b}=\left[\begin{array}{l}2 \\\3 \\\0 \\\5\end{array}\right]$$

Determine all values of \(k\) for which the given linear system has (a) no solution, (b) a unique solution, and (c) infinitely many solutions. $$\begin{array}{r} x_{1}-k x_{2}=6, \\ 2 x_{1}+3 x_{2}=k. \end{array}$$

Prove that (a) the inverse of an invertible upper triangular matrix is upper triangular. Repeat for an invertible lower triangular matrix. (b) the inverse of a unit upper triangular matrix is unit upper triangular. Repeat for a unit lower triangular matrix.

Determine all values of \(k\) for which the given linear system has (a) no solution, (b) a unique solution, and (c) infinitely many solutions. $$\begin{aligned} x_{1}-k x_{2}+k^{2} x_{3} &=0, \\ x_{1}+k x_{3} &=0, \\ x_{2}-x_{3} &=1. \end{aligned}$$

Prove that for each positive integer \(n\), there is a unique scalar matrix whose trace is a given constant \(k\) If \(A\) is an \(n \times n\) matrix, then the matrices \(B\) and \(C\) defined by $$ B=\frac{1}{2}\left(A+A^{T}\right), \quad C=\frac{1}{2}\left(A-A^{T}\right) $$ are referred to as the symmetric and skew-symmetric parts of \(A\) respectively. Problems \(32-36\) investigate properties of \(B\) and \(C\)

Show that an \(n \times n\) symmetric upper triangular matrix is diagonal. [Hint: This amounts to showing that if \(\left.i \neq j, \text { then } a_{i j}=0 .\right]\)

Use properties of the inverse to prove the given statement. If \(A\) is an \(n \times n\) invertible skew-symmetric matrix, then \(A^{-1}\) is skew-symmetric.