Exam 10: Systems and Matrices

Provide an appropriate response. -For any augmented matrix of a system of linear equations, which of the following row transformations will not necessarily result in a matrix of an equivalent system? (i) Adding to the elements of any row a multiple of the corresponding elements of another row. (ii) Interchanging any two columns. (iii) Multiplying the elements of any row by the corresponding elements of another row. (iv) Multiplying the elements of any row by any nonzero number.

Solve the system by elimination. - 6x+63=9y -3x+5y=36

Find the dimension of the matrix. - $\left[ \begin{array} { r r r } - 5 & 0 & 0 \\- 5 & - 5 & 0 \\5 & 3 & - 5\end{array} \right]$

Evaluate the determinant. - $\left| \begin{array} { l l l l } 0 & 8 & 1 & 6 \\0 & 9 & 4 & 2 \\3 & 2 & 2 & 7 \\2 & 3 & 1 & 3\end{array} \right|$

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - ax+by= x+y=

Graph the solution set of the system of inequalities. - x+2y\geq2 x-y\leq0

For the equation, determine the constants A and B that make the equation an identity. - $\frac { x - 4 } { x ^ { 2 } } = \frac { A } { x } + \frac { B } { x ^ { 2 } }$

Provide an appropriate response - 5x+3y=19 7x+8y=48 If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

Find the partial fraction decomposition for the rational expression. - $\frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 3 } { x ^ { 2 } + x - 2 }$

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - 3.2x+\piy= x-7.84y=4

Which method should be used to solve the system? Explain your answer, including a description of the first step. - -3+ =9 -8+9 =25

Find the matrix product when possible. - $\left[ \begin{array} { r r r } - 9 & - 5 & - 5 \\1 & - 1 & 8 \\9 & - 1 & 7\end{array} \right] \left[ \begin{array} { r r r } 4 & - 9 & 2 \\2 & 2 & - 2 \\- 3 & 9 & - 1\end{array} \right]$

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { c c } - 2 & 5 \\ 14 & 0 \end{array} \right] ; 7$ times row 1 added to row 2

Find the matrix product when possible. - $\left[ \begin{array} { r r r } 2 & - 4 & 7 \\7 & - 1 & - 3\end{array} \right] \left[ \begin{array} { r r r } 1 & - 4 & 3 \\2 & 8 & - 3 \\- 3 & 4 & - 4\end{array} \right]$

Provide an appropriate response. -Suppose that you are solving a system of 3 linear equations in 3 variables by the Gauss-Jordan method. If you use the transformation $- 10 \mathrm { R } _ { 3 } + \mathrm { R } _ { 2 }$ , which row or rows of the augmented matrix, if any, will change?

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - \pix+ey+z=2 ex+\piy+z=4 x+ey+\piz=6

Solve the problem. -Find the equation of the circle, written in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$ , passing through the points $( 12,2 ) , ( 7 , - 3 )$ , and $( 10,6 )$ .

Find the indicated matrix. -Let $B = \left[ \begin{array} { l l l l } - 1 & 3 & 6 & - 3 \end{array} \right]$ . Find $- 4 B$ .

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - -6x-7y=7 4x+5y=-5