Chapter 4

Write a matrix equation for each system of equations. \(x-y=-3\) \(x+3 y=5\)

Determine whether each pair of matrices are inverses of each other. $$ A=\left[\begin{array}{cc}{2} & {-1} \\ {1} & {-3}\end{array}\right], B=\left[\begin{array}{rr}{\frac{1}{2}} & {0} \\ {0} & {-\frac{1}{3}}\end{array}\right] $$

Use Cramer’s Rule to solve each system of equations. \(x-4 y=1\) \(2 x+3 y=13\)

Find the value of each determinant. $$ \left|\begin{array}{rr}{7} & {8} \\ {3} & {-2}\end{array}\right| $$

Triangle \(A B C\) with vertices \(A(1,4), B(2,-5),\) and \(C(-6,-6)\) is translated 3 units right and 1 unit down. Write the translation matrix.

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(A_{3 \times 5} \cdot B_{5 \times 2}\)

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{ccc}{5} & {8} & {-4}\end{array}\right]+\left[\begin{array}{cc}{12} & {5}\end{array}\right] $$

Write a matrix equation for each system of equations. \(2 g+3 h=8\) \(-4 g-7 h=-5\)

Determine whether each pair of matrices are inverses of each other. $$ X=\left[\begin{array}{ll}{3} & {1} \\ {5} & {2}\end{array}\right], Y=\left[\begin{array}{rr}{2} & {-1} \\ {-5} & {3}\end{array}\right] $$

Find the value of each determinant. $$ \left|\begin{array}{rr}{-3} & {-6} \\ {4} & {8}\end{array}\right| $$

Triangle \(A B C\) with vertices \(A(1,4), B(2,-5),\) and \(C(-6,-6)\) is translated 3 units right and 1 unit down. Find the coordinates of \(\triangle A^{\prime} B^{\prime} C^{\prime}\)

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(X_{2 \times 3} \cdot Y_{2 \times 3}\)

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{cc}{12} & {6} \\ {-8} & {-3}\end{array}\right]+\left[\begin{array}{cc}{14} & {-9} \\ {11} & {-6}\end{array}\right] $$

Determine whether each pair of matrices are inverses of each other. $$ C=\left[\begin{array}{rr}{1} & {-1} \\ {0} & {1}\end{array}\right], D=\left[\begin{array}{ll}{1} & {1} \\ {0} & {1}\end{array}\right] $$

Jarrod Wright has a total of \(\$ 5000\) in his savings account and in a certificate of deposit. His savings account earns 3.5\(\%\) interest annually. The certificate of deposit pays 5\(\%\) interest annually if the money is invested for one year. He calculates that his interest earnings for the year will be \(\$ 227.50\). Write a system of equations for the amount of money in each investment.

Evaluate each determinant using expansion by minors. $$ \left|\begin{array}{ccc}{0} & {-4} & {0} \\ {3} & {-2} & {5} \\ {2} & {-1} & {1}\end{array}\right| $$

Triangle \(A B C\) with vertices \(A(1,4), B(2,-5),\) and \(C(-6,-6)\) is translated 3 units right and 1 unit down. Graph the preimage and the image.

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(R_{3 \times 2} S_{2 \times 22}\)

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rr}{3} & {7} \\ {-2} & {1}\end{array}\right]-\left[\begin{array}{ll}{2} & {-3} \\ {5} & {-4}\end{array}\right] $$

State the dimensions of each matrix. $$ \left[\begin{array}{lllll}{3} & {4} & {5} & {6} & {7}\end{array}\right] $$

Use a matrix equation to solve each system of equations. \(5 x-3 y=-30\) \(8 x+5 y=1\)

Determine whether each pair of matrices are inverses of each other. $$ F=\left[\begin{array}{ll}{3} & {1} \\ {4} & {2}\end{array}\right], G=\left[\begin{array}{rr}{1} & {-2} \\ {-3} & {4}\end{array}\right] $$

Jarrod Wright has a total of \(\$ 5000\) in his savings account and in a certificate of deposit. His savings account earns 3.5\(\%\) interest annually. The certificate of deposit pays 5\(\%\) interest annually if the money is invested for one year. He calculates that his interest earnings for the year will be \(\$ 227.50\). How much money is in his savings account and in the certificate of deposit?

Evaluate each determinant using expansion by minors. $$ \left|\begin{array}{lll}{2} & {3} & {4} \\ {6} & {5} & {7} \\ {1} & {2} & {8}\end{array}\right| $$

Find each product, if possible. \(\left[\begin{array}{rr}{2} & {1} \\ {7} & {-5}\end{array}\right] \cdot\left[\begin{array}{cc}{-6} & {3} \\ {-2} & {-4}\end{array}\right]\)

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rr}{4} & {12} \\ {-3} & {-7}\end{array}\right]-\left[\begin{array}{rr}{5} & {3} \\ {-4} & {-4}\end{array}\right] $$

State the dimensions of each matrix. $$ \left[\begin{array}{rrrr}{10} & {-6} & {18} & {0} \\ {-7} & {5} & {2} & {4} \\\ {3} & {11} & {9} & {7}\end{array}\right] $$

Use a matrix equation to solve each system of equations. \(5 s+4 t=12\) \(4 s-3 t=-1.25\)

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{8} & {-5} \\ {-3} & {2}\end{array}\right] $$

Use Cramer’s Rule to solve each system of equations. \(2 x-y+3 z=5\) \(3 x+2 y-5 z=4\) \(x-4 y+11 z=3\)

Evaluate each determinant using diagonals. $$ \left|\begin{array}{rrr}{1} & {6} & {4} \\ {-2} & {3} & {1} \\ {1} & {6} & {4}\end{array}\right| $$

Find each product, if possible. \(\left[\begin{array}{rr}{10} & {-2} \\ {-7} & {3}\end{array}\right] \cdot\left[\begin{array}{rr}{1} & {4} \\ {5} & {-2}\end{array}\right]\)

Solve each equation. $$ \left[\begin{array}{c}{x+4} \\ {2 y}\end{array}\right]=\left[\begin{array}{c}{9} \\ {12}\end{array}\right] $$

Use a matrix equation to solve each system of equations. \(3 x+6 y=11\) \(2 x+4 y=7\)

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{4} & {-8} \\ {-1} & {2}\end{array}\right] $$

Use Cramer’s Rule to solve each system of equations. \(a+9 b-2 c=2\) \(-a-3 b+4 c=1\) \(2 a+3 b-6 c=-5\)

Find each product, if possible. \(\left[\begin{array}{ll}{3} & {-5}\end{array}\right] \cdot\left[\begin{array}{rr}{3} & {5} \\ {-2} & {0}\end{array}\right]\)

Solve each equation. $$ [9 \quad 13]=[x+2 y \quad 4 x+1] $$

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{-5} & {1} \\ {7} & {4}\end{array}\right] $$

Use Cramer’s Rule to solve each system of equations. \(5 x+2 y=8\) \(2 x-3 y=7\)

GEOMETRY What is the area of \(\triangle A B C\) with \(A(5,4), B(3,-4),\) and \(C(-3,-2) ?\)

Write a matrix equation for each system of equations. \(3 x-y=0\) \(x+2 y=-21\)

Code a message using your own coding matrix. Give your message and the matrix to a friend to decode. (Hint: Use coding matrix whose determinant is 1 and that has all positive elements.)

Use Cramer’s Rule to solve each system of equations. \(2 m+7 n=4\) \(m-2 n=-20\)

Find the area of the triangle whose vertices are located at \((2,-1),(1,2),\) and \((-1,0) .\)

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ 3\left[\begin{array}{rrrr}{6} & {-1} & {5} & {2} \\ {7} & {3} & {-2} & {8}\end{array}\right] $$

Write a matrix equation for each system of equations. \(4 x-7 y=2\) \(3 x+5 y=9\)

Determine whether each pair of matrices are inverses of each other. $$ P=\left[\begin{array}{ll}{0} & {1} \\ {1} & {1}\end{array}\right], Q=\left[\begin{array}{rr}{-1} & {1} \\ {1} & {0}\end{array}\right] $$

Use Cramer’s Rule to solve each system of equations. \(2 r-s=1\) \(3 r+2 s=19\)

Find the value of each determinant. $$ \left|\begin{array}{cc}{10} & {6} \\ {5} & {5}\end{array}\right| $$