Chapter 6

Evaluate the expression. $$ \left[\begin{array}{rr} 6 & 8 \\ -1 & 0 \end{array}\right]+\left[\begin{array}{rr} 0 & 5 \\ -3 & -1 \end{array}\right]+\left[\begin{array}{rr} -11 & -7 \\ 2 & -1 \end{array}\right] $$

Find the determinant of the matrix. $$ \left[\begin{array}{ll} -\frac{1}{2} & \frac{1}{3} \\ -6 & \frac{1}{3} \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rr} 11 & 1 \\ -1 & 0 \end{array}\right] $$

Evaluate the expression. $$ 4\left(\left[\begin{array}{rrr} -4 & 0 & 1 \\ 0 & 2 & 3 \end{array}\right]-\left[\begin{array}{rrr} 2 & 1 & -2 \\ 3 & -6 & 0 \end{array}\right]\right) $$

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrr} -2 & 5 & 1 \\ 3 & -1 & -8 \end{array}\right] \quad\left[\begin{array}{rrr} 13 & 0 & -39 \\ 3 & -1 & -8 \end{array}\right] $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} \frac{2}{3} & \frac{4}{3} \\ -1 & -\frac{1}{3} \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rr} -7 & 33 \\ 4 & -19 \end{array}\right] $$

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrr} 3 & -1 & -4 \\ -4 & 3 & 7 \end{array}\right] \quad\left[\begin{array}{rrr} 3 & -1 & -4 \\ 5 & 0 & -5 \end{array}\right] $$

A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure. From the northernmost vertex \(A\) of the region, the distances to the other vertices are 30 miles south and 15 miles east (for vertex \(B\) ), and 25 miles south and 33 miles east (for vertex \(C\) ). Use a graphing utility to approximate the number of square miles in this region.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 0.1 & 0.3 & 0.2 \\ -0.3 & -0.2 & 0.1 \\ 1 & 2 & 3 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rr} 8 & 4 \\ -2 & -2 \end{array}\right] $$

Evaluate the expression. $$ -3\left(\left[\begin{array}{rr} 0 & -3 \\ 7 & 2 \end{array}\right]+\left[\begin{array}{rr} -6 & 3 \\ 8 & 1 \end{array}\right]\right)-2\left[\begin{array}{ll} 4 & -4 \\ 7 & -9 \end{array}\right] $$

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrrr} 0 & -1 & -5 & 5 \\ -1 & 3 & -7 & 6 \\ 4 & -5 & 1 & 3 \end{array}\right] \quad\left[\begin{array}{rrrr} -1 & 3 & -7 & 6 \\ 0 & -1 & -5 & 5 \\ 0 & 7 & -27 & 27 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 0.2 & -0.1 & -0.3 \\ 0.1 & -0.1 & 0.4 \\ -0.5 & -0.2 & -0.1 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right] $$

Evaluate the expression. $$ -1\left[\begin{array}{rr} 4 & 11 \\ -2 & -1 \\ 9 & 3 \end{array}\right]+\frac{1}{6}\left(\left[\begin{array}{rr} -5 & -1 \\ 3 & 4 \\ 0 & 13 \end{array}\right]+\left[\begin{array}{rr} 7 & 5 \\ -9 & -1 \\ 6 & -1 \end{array}\right]\right) $$

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrrr} -1 & -2 & 3 & -2 \\ 2 & -5 & 1 & -7 \\ 5 & 4 & -7 & 6 \end{array}\right] \quad\left[\begin{array}{rrrr} -1 & -2 & 3 & -2 \\ 0 & -9 & 7 & -11 \\ 0 & -6 & 8 & -4 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (-4,-7),(0,-4),(4,-1) $$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 0.9 & 0.7 & 0 \\ -0.1 & 0.3 & 1.3 \\ 2.2 & 4.2 & 6.1 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rr} 0 & 4 \\ -3 & 6 \end{array}\right] $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (2,4),(4,5),(-2,2) $$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 0.1 & 0.1 & -4.3 \\ 7.5 & 6.2 & 0.7 \\ 0.3 & 0.6 & -1.2 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{ll} 2 & 3 \\ 6 & 9 \end{array}\right] $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 1 & -3 & 10 \\ 0 & 0 & 1 & 0 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (-1,-7),(0,-3),(1,2) $$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{rrr} 5 & -3 & 2 \\ 7 & 5 & -7 \\ 0 & 6 & -1 \end{array}\right] $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 2 & 0 & 4 & 0 \\ 0 & -1 & 3 & 6 \\ 0 & 0 & 1 & 5 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (1,7),(0,4),(-1,2) $$

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$ \left[\begin{array}{llr} 2 & 3 & 1 \\ 0 & 5 & -2 \\ 0 & 0 & -2 \end{array}\right] $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 3 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (-2,-11),(4,13),(2,5) $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rr} 3 & 4 \\ 2 & -5 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 5 & 4 \\ 3 & 6 & 5 \end{array}\right] $$

Solve for \(X\) when $$A=\left[\begin{array}{rr} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right].$$ $$ X=3 \dot{A}-2 \bar{B} $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{llllll} \mathbf{1} & \underline{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 8 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (4,3),(3,1),(2,-1) $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rr} 11 & 0 \\ -3 & 2 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 1 & 2 & 2 \\ 3 & 7 & 9 \\ -1 & -4 & -7 \end{array}\right] $$

Solve for \(X\) when $$A=\left[\begin{array}{rr} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right].$$ $$ 2 X=2 A-B $$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$ \left[\begin{array}{llll} 1 & 0 & 0 & 10 \\ 0 & 1 & 3 & 9 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Use a determinant to determine whether the points are collinear. $$ (-2,3),(2,-1),(7,-4) $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rr} 3 & 1 \\ -2 & -4 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 1 & 1 & 2 \\ 3 & 1 & 0 \\ -2 & 0 & 3 \end{array}\right] $$

Solve for \(X\) when $$A=\left[\begin{array}{rr} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right].$$ $$ 2 X+3 A=B $$

Use a graphing utility to perform the sequence of row operations in parts (a) through (d) to reduce the matrix to row-echelon form. \(\left[\begin{array}{rrr}1 & 1 & 2 \\ 3 & 4 & -3 \\ 2 & -1 & 1\end{array}\right]\) (a) Add \(-3\) times \(R_{1}\) to \(R_{2}\). (b) Add \(-2\) times \(R_{1}\) to \(R_{3}\). (c) Add 3 times \(R_{2}\) to \(R_{3}\). (d) Multiply \(R_{3}\) by \(-\frac{1}{30}\).

Use a determinant to determine whether the points are collinear. $$ (-3,-4),(-1,-1),(5,5) $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rr} -6 & 5 \\ 7 & -2 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 3 & 2 & 2 \\ 2 & 2 & 2 \\ -4 & 4 & 3 \end{array}\right] $$

Solve for \(X\) when $$A=\left[\begin{array}{rr} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right].$$ $$ 2 A+4 B=-2 X $$