Exam 6: Matrices and Determinants

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\0 & 1 & - 2 & 0 \\0 & 0 & 1 & 9 \\0 & 0 & 0 & 1\end{array} \right]$

Evaluate the determinant. - $\left| \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 5 & 5 \\ 1 & 2 & 3 \end{array} \right|$

Evaluate the determinant. - $\left| \begin{array} { c c } \frac { 1 } { 6 } & \frac { 1 } { 4 } \\ - \frac { 7 } { 9 } & \frac { 11 } { 5 } \end{array} \right|$

Evaluate the determinant. - $\left| \begin{array} { l l } 2 & 2 \\ 1 & 3 \end{array} \right|$

Write a system of linear equations in three variables, and then use matrices to solve the system. -A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 15 hours to prepare, 3 hours to paint, and 4 Hours to fire. A sleigh takes 4 hours to prepare, 16 hours to paint, and 7 hours to fire. If the workshop has 93 hours for prep time, 74 hours for painting, and 107 hours for firing, how many of each can be made?

Perform the matrix row operation (or operations)and write the new matrix. - $\left[ \begin{array} { r r r | r } 1 & - 4 & 1 & 3 \\ - 5 & 0 & 3 & - 3 \\ - 1 & 2 & - 2 & - 1 \end{array} \right] - 3 R _ { 1 } + R _ { 2 }$

Use Matrices and Gauss-Jordan Elimination to Solve Systems Solve the system of equations using matrices. Use Gauss-Jordan elimination. - x=6-y-z x-y+3z=-6 2x+y=8-z

Evaluate the determinant. - $\left| \begin{array} { l l l } 2 & 1 & 5 \\ 2 & 6 & 5 \\ 1 & 5 & 5 \end{array} \right|$

Find the product AB, if possible. - $\mathrm { A } = \left[ \begin{array} { r r } - 2 & 3 \\ 3 & 2 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l } - 2 & 0 \\ - 1 & 3 \end{array} \right]$

Use Inverses to Solve Matrix Equations Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. - $\left[ \begin{array} { r r } 8 & 2 \\3 & - 6\end{array} \right] \left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } 6 \\9\end{array} \right]$

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $\mathrm { A } = \left[ \begin{array} { r r } 4 & - 3 \\ - 4 & 0 \end{array} \right]$

Encode and Decode Messages Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order. -Use the coding matrix $A = \left[ \begin{array} { l l } 3 & 7 \\ 2 & 5 \end{array} \right]$ to encode the message LIFE.