91Ó°ÊÓ

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{rrr} 3 & 7 & 10 \\ 2 & 3 & -1 \\ 1 & 2 & 1 \end{array}\right]$$.

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrrr}2 & -3 & 1 & 2 \\\4 & -1 & 1 & 1 \\\\-8 & 2 & 2 & -5 \\\6 & 1 & 5 & 2 \end{array}\right]$$

Determine the solution set to the given linear system of equations. $$\begin{aligned} 5 x_{1}-x_{2}+2 x_{3} &=7, \\ -2 x_{1}+6 x_{2}+9 x_{3} &=0, \\ -7 x_{1}+5 x_{2}-3 x_{3} &=-7. \end{aligned}$$

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{ccc} 3 & -3 & 6 \\ 2 & -2 & 4 \\ 6 & -6 & 12 \end{array}\right]$$.

Use the LU factorization of \(A\) to solve the system \(A \mathbf{x}=\mathbf{b}\). $$A=\left[\begin{array}{ll}1 & 2 \\\2 & 3\end{array}\right], \mathbf{b}=\left[\begin{array}{r}3 \\\\-1 \end{array}\right]$$

Determine the solution set to the given linear system of equations. $$\begin{aligned} x+2 y-z =1, \\ x+z=5, \\ 4 x+4 y =12. \end{aligned}$$

Determine the solution set to the sys\(\operatorname{tem} A \mathbf{x}=\mathbf{b}\) for the given coefficient matrix \(A\) and right-hand side vector b. $$A=\left[\begin{array}{rrrr} 1 & -1 & 0 & -1 \\ 2 & 1 & 3 & 7 \\ 3 & -2 & 1 & 0 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 2 \\ 2 \\ 4 \end{array}\right]$$

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

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

Determine the solution set to the sys\(\operatorname{tem} A \mathbf{x}=\mathbf{b}\) for the given coefficient matrix \(A\) and right-hand side vector b. $$A=\left[\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ 3 & 1 & -2 & 3 \\ 2 & 3 & 1 & 2 \\ -2 & 3 & 5 & -2 \end{array}\right], \mathbf{b}=\left[\begin{array}{r} 2 \\ 8 \\ 3 \\ -9 \end{array}\right]$$