Chapter 6

Find the determinant of the matrix. $$ [-5] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{ll}7 & 4 \\ 5 & 3\end{array}\right], B=\left[\begin{array}{rr}3 & -4 \\ -5 & 7\end{array}\right]\)

Find \(x\) and \(y\). $$ \left[\begin{array}{rr} 4 & x \\ -1 & y \end{array}\right]=\left[\begin{array}{rr} 4 & -3 \\ -1 & 2 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{rrr} 0 & -3 & 0 \\ 9 & 2 & -7 \end{array}\right] $$

Find the determinant of the matrix. $$ [6] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{ll}-4 & 1 \\ -9 & 2\end{array}\right], B=\left[\begin{array}{ll}2 & -1 \\ 9 & -4\end{array}\right]\)

Find \(x\) and \(y\). $$ \left[\begin{array}{rr} x & -7 \\ 9 & y \end{array}\right]=\left[\begin{array}{ll} 5 & -7 \\ 9 & -8 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{ll} -7 & 21 \end{array}\right] $$

Find the determinant of the matrix. $$ \left[\begin{array}{ll} 1 & 3 \\ 2 & 7 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{ll}2 & -1 \\ 5 & -4\end{array}\right], B=\left[\begin{array}{ll}\frac{4}{3} & -\frac{1}{3} \\ \frac{5}{3} & -\frac{2}{3}\end{array}\right]\)

Find \(x\) and \(y\). $$ \left[\begin{array}{rr} -4 & 3 \\ 6 & -1 \\ 8 & 2 \\ 5 & 9 \end{array}\right]=\left[\begin{array}{cc} x-2 & 3 \\ 6 & -1 \\ 8 & -x \\ 5 & 2 y-1 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{rrr} 6 & 4 & 1 \\ 8 & 3 & 0 \\ -1 & 2 & 1 \\ 1 & 5 & 4 \end{array}\right] $$

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

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rr}1 & -2 \\ 3 & -10\end{array}\right], B=\left[\begin{array}{ll}\frac{5}{2} & -\frac{1}{2} \\ \frac{3}{4} & -\frac{1}{4}\end{array}\right]\)

Find \(x\) and \(y\). $$ \left[\begin{array}{ccc} x+2 & 8 & -3 \\ 1 & 2 y & 2 x \\ 7 & -2 & y+2 \end{array}\right]=\left[\begin{array}{ccc} 2 x+6 & 8 & -3 \\ 1 & 18 & -8 \\ 7 & -2 & 11 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{l} 1 \\ 0 \\ 3 \\ 5 \\ 6 \end{array}\right] $$

Find the determinant of the matrix. $$ \left[\begin{array}{ll} 5 & 6 \\ 2 & 3 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}-2 & 2 & 3 \\ 1 & -1 & 0 \\ 0 & 1 & 4\end{array}\right], B=\frac{1}{3}\left[\begin{array}{rrr}-4 & -5 & 3 \\ -4 & -8 & 3 \\ 1 & 2 & 0\end{array}\right]\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ A=\left[\begin{array}{rr} 5 & -2 \\ 3 & 1 \end{array}\right], B=\left[\begin{array}{rr} 3 & 1 \\ -2 & 6 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{rr} 33 & 45 \\ -9 & 20 \\ 12 & 15 \\ 16 & -2 \end{array}\right] $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} -7 & -4 \\ 8 & 7 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}-1 & 0 & 2 \\ 1 & -2 & 0 \\ 1 & 0 & 3\end{array}\right], B=\frac{1}{10}\left[\begin{array}{rrr}-6 & 0 & 4 \\ -3 & -5 & 2 \\ 2 & 0 & 2\end{array}\right]\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ A=\left[\begin{array}{rr} 7 & 4 \\ -4 & 5 \end{array}\right], B=\left[\begin{array}{rr} -3 & 1 \\ 8 & -4 \end{array}\right] $$

Determine the order of the matrix. $$ \left[\begin{array}{rrr} 12 & -2 & 4 \\ -3 & 4 & 0 \\ -8 & 12 & 2 \end{array}\right] $$

Use a determinant to find the area of the triangle with the given vertices. $$ (-2,4),(2,3),(-1,5) $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} 9 & 3 \\ 12 & 4 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}2 & -17 & 11 \\ -1 & 11 & -7 \\ 0 & 3 & -2\end{array}\right], B=\left[\begin{array}{llr}1 & 1 & 2 \\ 2 & 4 & -3 \\ 3 & 6 & -5\end{array}\right]\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ A=\left|\begin{array}{rr} 6 & -1 \\ 2 & 4 \\ -3 & 5 \end{array}\right|, B=\left|\begin{array}{rr} 1 & 4 \\ -1 & 5 \\ 1 & 10 \end{array}\right| $$

Determine the order of the matrix. $$ \left[\begin{array}{rrrr} 2 & 7 & 11 & -3 \\ -1 & 10 & -5 & 0 \end{array}\right] $$

Use a determinant to find the area of the triangle with the given vertices. $$ (0,-2),(-1,4),(3,5) $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} -5 & -2 \\ 10 & 4 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}-1 & 1 & -3 \\ 2 & -1 & 4 \\ -1 & 1 & -2\end{array}\right], B=\left[\begin{array}{rrr}2 & 1 & -1 \\ 0 & 1 & 2 \\\ -1 & 0 & 1\end{array}\right]\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ \begin{aligned} &A=\left[\begin{array}{rrrrr} 6 & 8 & -3 & 2 & 1 \\ -4 & 2 & 1 & 5 & -2 \end{array}\right], \\ &B=\left[\begin{array}{llrrr} 6 & 0 & 4 & -1 & 3 \\ 4 & 5 & -2 & 1 & 2 \end{array}\right] \end{aligned} $$

Determine the order of the matrix. $$ [-11] $$

Use a determinant to find the area of the triangle with the given vertices. $$ (-3,5),(2,6),(3,-5) $$

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

Show that \(B\) is the inverse of \(A\). \(\begin{aligned} A &=\left[\begin{array}{rrrr}2 & 0 & 2 & 1 \\ 3 & 0 & 0 & 1 \\\ -1 & 1 & -2 & 1 \\ 3 & -1 & 1 & 0\end{array}\right] \\ B &=\frac{1}{3}\left[\begin{array}{rrrr}-1 & 3 & -2 & -2 \\ -2 & 9 & -7 & -10 \\\ 1 & 0 & -1 & -1 \\ 3 & -6 & 6 & 6\end{array}\right] \end{aligned}\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ \mathbf{A}=\left[\begin{array}{rrr} 2 & 2 & -1 \\ 1 & 1 & -2 \\ 1 & -1 & 3 \end{array}\right], B=\left[\begin{array}{rrr} 1 & 1 & -1 \\ -3 & 4 & 9 \\ 0 & -7 & 8 \end{array}\right] $$

Use a determinant to find the area of the triangle with the given vertices. $$ (-2,4),(1,5),(3,-2) $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} -3 & 1 \\ 5 & 2 \end{array}\right] $$

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrrr}-1 & 1 & 0 & -1 \\ 1 & -1 & 2 & 0 \\ -1 & 1 & 2 & 0 \\ 0 & -1 & 1 & 1\end{array}\right]\) \(B=\frac{1}{4}\left[\begin{array}{rrrr}-4 & 1 & 1 & -4 \\ -4 & -1 & 3 & -4 \\\ 0 & 1 & 1 & 0 \\ -4 & -2 & 2 & 0\end{array}\right]\)

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ A=\left[\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right], B=\left[\begin{array}{r} -4 \\ 6 \\ 2 \end{array}\right] $$

In Exercises 11 and 12 , find a value of \(y\) such that the triangle with the given vertices has an area of 4 square units. $$ (-5,1),(0,2),(-2, y) $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} \frac{2}{3} & 0 \\ -1 & 6 \end{array}\right] $$

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

Evaluate the expression. $$ \left[\begin{array}{rr} -5 & 0 \\ 3 & -6 \end{array}\right]+\left[\begin{array}{rr} 7 & 1 \\ -2 & -1 \end{array}\right]+\left[\begin{array}{rr} -10 & -8 \\ 14 & 6 \end{array}\right] $$

Fill in the blank(s) to form a new row-equivalent matrix. Original Matrix New Row-Reduced Matrix .\(\left[\begin{array}{rrrr}1 & 5 & 4 & -1 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 1 & -7\end{array}\right] \quad\left[\begin{array}{rrrr}1 & 0 & & \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 1 & -7\end{array}\right]\)

In Exercises 11 and 12 , find a value of \(y\) such that the triangle with the given vertices has an area of 4 square units. $$ (-4,2),(-3,5),(-1, y) $$

Find the determinant of the matrix. $$ \left[\begin{array}{rr} 9 & -\frac{1}{4} \\ 8 & 0 \end{array}\right] $$

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