Exam 6: Matrices and Determinants and Applications

Perform the elementary row operation on the given matrix. - $- \frac { 1 } { 2 } R _ { 2 } \rightarrow R _ { 2 }$ $\left[ \begin{array} { r r | r } 9 & 11 & 1 \\16 & 32 & 20\end{array} \right]$

Write a system of linear equations represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 4 & 9 & - 4 & - 12 \\9 & 2 & 7 & 6 \\4 & - 2 & 2 & - 1\end{array} \right]$

Evaluate the determinant of the given matrix and state whether the matrix is invertible. - $A = \left[ \begin{array} { l l l } 7 & 4 & 5 \\5 & - 5 & 8 \\1 & 5 & 1\end{array} \right]$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. - $I _ { n }$ is an $n \times n$ matrix with ________'s along the main diagonal and ________'s elsewhere.

Explain the meaning of the notation $R _ { 2 } \Longleftrightarrow R _ { 3 }$

Determine if the matrix is in row-echelon form. - $\left[ \begin{array} { r r r | r } 1 & - 6 & - 8 & - 7 \\0 & 1 & 0 & 8 \\0 & 0 & 2 & 3\end{array} \right]$

Using the determinant $\left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\x _ { 2 } & y _ { 2 } & 1 \\x _ { 3 } & y _ { 3 } & 1\end{array} \right|$ , determine if the points are collinear. - $( 8,9 ) , ( 2,2 ) , ( - 4 , - 5 )$

Solve the system, if possible, by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. - 6x-9y+9z=3 -4x+6y-6z=-2 2x-3y+3z=2

Explain the meaning of the notation $3 \mathrm { R } _ { 1 } \rightarrow \mathrm { R } _ { 1 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Suppose that the given system has one solution. x+y= x+y= Cramer's rule gives the solution as $x = \frac { \square } { \square }$ and $y = \frac { \square } { \square }$ where $D = \left| \begin{array} { l } \square \square \\ \square \square \end{array} \right| , D _ { x } = \left| \begin{array} { l } \square \square \\ \square \square \end{array} \right|$ , and $D_{y}=\left|\begin{array}{l}\square \square \\\square \square\end{array}\right|$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -To multiply two matrices A and B, the number of ________of A must equal the number of________of B.

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - -5x+9y-9z=-106 2x-2y-5z=-14 -5x-8y+2z=11

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -True or false: Matrix multiplication is a commutative operation.

Write the system of equations as a matrix equation of the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants. - -5x-2y=-7 3x+3y=6

Find the minor of the given element. - A= = -2 4 -2 8 1 -9 -2 9 7

Choose the one alternative that best completes the statement or answers the question. Verify $A I _ { n } = A$ - $A = \left[ \begin{array} { r r r } - 9 & 4 & 6 \\- 6 & 3 & - 1 \\- 9 & - 3 & 5\end{array} \right]$

Choose the one alternative that best completes the statement or answers the question. Write the augmented matrix for the given system. - x + 3 y + 5 z = - 3 - 5 y + 3 z = - 8 - 8 z = - 6

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -When applying Cramer's rule, if D = 0, and all other determinants $D _ { i }$ in the numerator are zero, then the system as (no solution / infinitely many) solutions.

Determine the value of the given element of the matrix. - A= 3 -8 9 -2 15 -3 10 8 -7 2 -5 13

Solve the system by using the inverse of the coefficient matrix. - r-3s+t=8 -r+6s+t=3 3r-3s-t=6