Chapter 6

Use a graphing utility to perform the sequence of row operations in parts (a) through (f) to reduce the matrix to reduced row-echelon form. \(\left[\begin{array}{rr}7 & 1 \\ 0 & 2 \\ -3 & 4 \\ 4 & 1\end{array}\right]\) (a) \(\operatorname{Add} R_{3}\) to \(R_{4}\). (b) Interchange \(R_{1}\) and \(R_{4}\). (c) Add 3 times \(R_{1}\) to \(R_{3}\). (d) Add \(-7\) times \(R_{1}\) to \(R_{4}\). (e) Multiply \(R_{2}\) by \(\frac{1}{2}\). (f) Add the appropriate multiple of \(R_{2}\) to \(R_{1}, R_{3}\), and \(R_{4}\).

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} 4 & 0 & 2 \\ -3 & 2 & 1 \\ 1 & -1 & 1 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 4 \end{array}\right] $$

Find \(A B\), if possible. $$ A=\left[\begin{array}{rr} 3 & -2 \\ 4 & 5 \\ 1 & -1 \end{array}\right], B=\left[\begin{array}{rrrr} -1 & 4 & -2 & 5 \\ 2 & 1 & 3 & -1 \end{array}\right] $$

Write the matrix in row-echelon form. (Note: Row-echelon forms are not unique.) $$ \left[\begin{array}{rrrr} 1 & 2 & -1 & 5 \\ 3 & 2 & 1 & 11 \\ 4 & 8 & 1 & 10 \end{array}\right] $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} 1 & -1 & 0 \\ 3 & 2 & 5 \\ 4 & -6 & 4 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Find \(A B\), if possible. $$ A=\left[\begin{array}{rrr} 0 & -1 & 0 \\ 4 & 0 & 2 \\ 8 & -1 & 7 \end{array}\right], B=\left[\begin{array}{rr} 2 & 1 \\ -3 & 4 \\ 1 & 6 \end{array}\right] $$

Write the matrix in row-echelon form. (Note: Row-echelon forms are not unique.) $$ \left[\begin{array}{rrrr} 1 & 2 & -1 & 3 \\ 3 & 7 & -5 & 14 \\ -2 & -1 & -3 & 8 \end{array}\right] $$

In Exercises 27 and 28, find \(x\) such that the points are collinear. $$ (-4,-1),(-1,2),(x, 6) $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} 3 & -2 & 8 \\ 3 & 2 & -6 \\ -1 & 3 & 6 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 0 & 0 \\ 3 & 4 & 0 \\ 2 & 5 & 5 \end{array}\right] $$

Find \(A B\), if possible. $$ A=\left[\begin{array}{rr} -1 & 3 \\ 4 & -5 \\ 0 & 2 \end{array}\right], B=\left[\begin{array}{ll} 1 & 2 \\ 0 & 7 \end{array}\right] $$

Write the matrix in row-echelon form. (Note: Row-echelon forms are not unique.) $$ \left[\begin{array}{rrrr} 1 & -1 & -1 & 1 \\ 5 & -4 & 1 & 8 \\ -6 & 8 & 18 & 0 \end{array}\right] $$

Find all (a) minors and (b) cofactors of the matrix. $$ \left[\begin{array}{rrr} -2 & 9 & 4 \\ 7 & -6 & 0 \\ 6 & 7 & -6 \end{array}\right] $$

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 0 & 0 \\ 3 & 0 & 0 \\ 2 & 5 & 5 \end{array}\right] $$

Find \(A B\), if possible. $$ A=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -2 \end{array}\right], B=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Write the matrix in row-echelon form. (Note: Row-echelon forms are not unique.) $$ \left[\begin{array}{rrrr} 1 & -3 & 0 & -7 \\ -3 & 10 & 1 & 23 \\ 1 & 0 & 1 & 12 \\ 4 & -10 & 2 & -24 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ (-1,2),(5,3) $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrr}4 & 1 & -3 \\ 6 & 5 & -2 \\ -1 & 3 & -4\end{array}\right]\) (a) Row 3 (b) Column 2

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{llll} 1 & 0 & 3 & 0 \\ 0 & 2 & 0 & 4 \\ 1 & 0 & 3 & 0 \\ 0 & 2 & 0 & 4 \end{array}\right] $$

Find \(A B\), if possible. $$ A=\left[\begin{array}{rrr} 5 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 7 \end{array}\right], B=\left[\begin{array}{rrr} \frac{1}{5} & 0 & 0 \\ 0 & -\frac{1}{8} & 0 \\ 0 & 0 & \frac{1}{2} \end{array}\right] $$

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrr} 4 & 4 & 8 \\ 1 & 2 & 2 \\ -3 & 6 & -9 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ (3,1),(-2,-5) $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrr}-3 & 4 & 2 \\ 6 & 3 & 1 \\ 4 & -7 & -8\end{array}\right]\) (a) Row 2 (b) Column 3

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

Find \(A B\), if possible. $$ A=\left[\begin{array}{r} 6 \\ -2 \\ 1 \\ 6 \end{array}\right], B=\left[\begin{array}{ll} 10 & 12 \end{array}\right] $$

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrr} 1 & 3 & 2 \\ 5 & 15 & 9 \\ 2 & 6 & 10 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ (-4,3),(2,1) $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrr}7 & 0 & -4 \\ 2 & -3 & 0 \\ 5 & 8 & 1\end{array}\right]\) (a) Row 1 (b) Column 3

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 1 & 2 & -1 \\ 3 & 7 & -10 \\ -5 & -7 & -15 \end{array}\right] $$

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 3 & -6 & 6 \\ 2 & 0 & 5 & -4 \\ 0 & 1 & 0 & 1 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ (10,7),(-2,-7) $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrr}10 & -5 & 5 \\ 30 & 0 & 10 \\ 0 & 10 & 1\end{array}\right]\) (a) Row 3 (b) Column 1

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 10 & 5 & -7 \\ -5 & 1 & 4 \\ 3 & 2 & -2 \end{array}\right] $$

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 1 & 2 & 3 & -5 \\ 1 & 2 & 4 & -9 \\ -2 & -4 & -4 & 3 \\ 4 & 8 & 11 & -14 \end{array}\right] $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrrr}6 & 0 & -3 & 5 \\ 4 & 13 & 6 & -8 \\ -1 & 0 & 7 & 4 \\ 8 & 6 & 0 & 2\end{array}\right]\) (a) Row 2 (b) Column 2

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

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rr} 2 & 1 \\ 1 & 4 \\ -2 & -1 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ (3,3),(6,3) $$

Find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column. \(\left[\begin{array}{rrrr}10 & 8 & 3 & -7 \\ 4 & 0 & 5 & -6 \\ 0 & 3 & 2 & 7 \\\ 1 & 0 & -3 & 2\end{array}\right]\) (a) Row 3 (b) Column 1

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 0.6 & 0 & -0.3 \\ 0.7 & -1 & 0.2 \\ 1 & 0 & -0.9 \end{array}\right] $$

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rr} 1 & -3 \\ -1 & 8 \\ 0 & 4 \\ -2 & 10 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ \left(-\frac{1}{2}, 3\right),\left(\frac{5}{2}, 1\right) $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 2 & 0 \\ -1 & 4 & 3 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 1 & -3 & 2 & -1 \\ 0 & 4 & -12 & 8 \\ 3 & 0 & 5 & -2 \\ 0 & -3 & 9 & -6 \end{array}\right] $$

Use a determinant to find an equation of the line passing through the points. $$ \left(\frac{2}{3}, 4\right),(6,12) $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 2 & -1 & 3 \\ 1 & 4 & 4 \\ 1 & 0 & 2 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 1 & 3 & -2 & 0 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & -2 & 1 \\ 0 & 0 & 0 & 5 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 2 & 4 & 6 \\ 0 & 3 & 1 \\ 0 & 0 & -5 \end{array}\right] $$