91Ó°ÊÓ

You have a system of \(k\) equations in two variables, \(k \geq 2 .\) Explain the geometric significance of (a) No solution. (b) A unique solution. (c) An infinite number of solutions.

Do the three lines, \(x+2 y=1,2 x-y=1,\) and \(4 x+3 y=3\) have a common point of intersection? If so, find the point and if not, tell why they don't have such a common point of intersection.

Suppose a system of equations has fewer equations than variables. Will such a system necessarily be consistent? If so, explain why and if not, give an example which is not consistent.

Find h such that $$ \left[\begin{array}{ll|l} 2 & h & 4 \\ 3 & 6 & 7 \end{array}\right] $$ is the augmented matrix of an inconsistent system.

Choose h and k such that the augmented matrix shown has each of the following: (a) one solution (b) no solution (c) infinitely many solutions $$ \left[\begin{array}{ll|l} 1 & h & 2 \\ 2 & 4 & k \end{array}\right] $$

Row reduce the following matrix to obtain the row-echelon form. Then continue to obtain the reduced row-echelon form. $$ \left[\begin{array}{rrrr} 3 & -6 & -7 & -8 \\ 1 & -2 & -2 & -2 \\ 1 & -2 & -3 & -4 \end{array}\right] $$

Suppose a system of linear equations has a \(2 \times 4\) augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent? Explain.

Suppose the coefficient matrix of a system of n equations with \(n\) variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.

Find the rank of the following matrix. $$ \left[\begin{array}{rrrr} 4 & -16 & -1 & -5 \\ 1 & -4 & 0 & -1 \\ 1 & -4 & -1 & -2 \end{array}\right] $$

Suppose A is an \(m \times n\) matrix. Explain why the rank of \(A\) is always no larger than \(\min (m, n)\)