Matrices and Matrix Operations

1. No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a [latex]2\times 2[/latex] matrix and the second is a [latex]2\times 3[/latex] matrix. [latex]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right][/latex] has no sum.

2. No. Matrices can only be multiplied if the inner dimensions are equal. For example, a 2×3 matrix cannot be multiplied by another 2×3 matrix since 3 does not equal 2.

3. [latex]\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right][/latex]

4. [latex]\left[\begin{array}{cc}-4& 2\\ 8& 1\end{array}\right][/latex]

5. Undidentified; dimensions do not match

6. [latex]\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right][/latex]

7. [latex]\left[\begin{array}{cccc}-64& -12& -28& -72\\ -360& -20& -12& -116\end{array}\right][/latex]

8. [latex]\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right][/latex]

9. [latex]\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right][/latex]

10. [latex]\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right][/latex]

11. [latex]\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right][/latex]

12. Undefined; dimensions do not match.

13. [latex]\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right][/latex]

14. [latex]\left[\begin{array}{ccc}-840& 650& -530\\ 330& 360& 250\\ -10& 900& 110\end{array}\right][/latex]

15. [latex]\left[\begin{array}{cc}-350& 1,050\\ 350& 350\end{array}\right][/latex]

16. Undefined; inner dimensions do not match.

17. [latex]\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right][/latex]

18. [latex]\left[\begin{array}{cc}332,500& 927,500\\ -227,500& 87,500\end{array}\right][/latex]

19. [latex]\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right][/latex]

20. [latex]\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right][/latex]

21. [latex]\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right][/latex]

22. [latex]\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right][/latex]

23. [latex]\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right][/latex]

24. [latex]\left[\begin{array}{cc}0& 1.6\\ 9& -1\end{array}\right][/latex]

25. [latex]\left[\begin{array}{ccc}2& 24& -4.5\\ 12& 32& -9\\ -8& 64& 61\end{array}\right][/latex]

Solving Systems with Gaussian Elimination

1. Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.

2. [latex]\left[\left.\begin{array}{rrrr}\hfill 0& \hfill & \hfill 16& \hfill \\ \hfill 9& \hfill & \hfill -1& \hfill \end{array}\right\rvert\begin{array}{rr}\hfill & \hfill 4\\ \hfill & \hfill 2\end{array}\right][/latex]

3. [latex]\left[\left.\begin{array}{rrrrrr}\hfill 1& \hfill & \hfill 5& \hfill & \hfill 8& \hfill \\ \hfill 12& \hfill & \hfill 3& \hfill & \hfill 0& \hfill \\ \hfill 3& \hfill & \hfill 4& \hfill & \hfill 9& \hfill \end{array}\right\rvert\begin{array}{rr}\hfill & \hfill 16\\ \hfill & \hfill 4\\ \hfill & \hfill -7\end{array}\right][/latex]

4. [latex]\begin{array}{l}-2x+5y=5\\ 6x - 18y=26\end{array}[/latex]

5. [latex]\begin{array}{l}3x+2y=13\\ -x - 9y+4z=53\\ 8x+5y+7z=80\end{array}[/latex]

6. [latex]\begin{array}{l}4x+5y - 2z=12\hfill \\ \text{ }y+58z=2\hfill \\ 8x+7y - 3z=-5\hfill \end{array}[/latex]

7. No solutions

8. [latex]\left(-1,-2\right)[/latex]

9. [latex]\left(6,7\right)[/latex]

10. [latex]\left(\frac{1}{5},\frac{1}{2}\right)[/latex]

11. [latex]\left(x,\frac{4}{15}\left(5x+1\right)\right)[/latex]

12. [latex]\left(\frac{196}{39},-\frac{5}{13}\right)[/latex]

13. [latex]\left(31,-42,87\right)[/latex]

14. [latex]\left(\frac{21}{40},\frac{1}{20},\frac{9}{8}\right)[/latex]

15. [latex]\left(\frac{18}{13},\frac{15}{13},-\frac{15}{13}\right)[/latex]

16. [latex]\left(x,y,\frac{1}{2}\left(1 - 2x - 3y\right)\right)[/latex]

17. [latex]\left(125,-25,0\right)[/latex]

18. [latex]\left(8,1,-2\right)[/latex]

19. 860 red velvet, 1,340 chocolate

20. 4% for account 1, 6% for account 2

21. $126

22. Banana was 3%, pumpkin was 7%, and rocky road was 2%

23. 100 almonds, 200 cashews, 600 pistachios

Solving Systems with Inverses

1. No, because [latex]ad[/latex] and [latex]bc[/latex] are both 0, so [latex]ad-bc=0[/latex], which requires us to divide by 0 in the formula.

2. [latex]AB=BA=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I[/latex]

3. [latex]AB=BA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I[/latex]

4. [latex]\frac{1}{29}\left[\begin{array}{cc}9& 2\\ -1& 3\end{array}\right][/latex]

5. There is no inverse

6. [latex]\frac{4}{7}\left[\begin{array}{cc}0.5& 1.5\\ 1& -0.5\end{array}\right][/latex]

7. [latex]\frac{1}{17}\left[\begin{array}{ccc}-5& 5& -3\\ 20& -3& 12\\ 1& -1& 4\end{array}\right][/latex]

8. [latex]\left[\begin{array}{ccc}18& 60& -168\\ -56& -140& 448\\ 40& 80& -280\end{array}\right][/latex]

9. [latex]\left(-5,6\right)[/latex]

10. [latex]\left(\frac{1}{3},-\frac{5}{2}\right)[/latex]

11. [latex]\left(5,0,-1\right)[/latex]

12. [latex]\frac{1}{690}\left(65,-1136,-229\right)[/latex]

13. [latex]\left(-\frac{37}{30},\frac{8}{15}\right)[/latex]

14. [latex]\left(\frac{10}{123},-1,\frac{2}{5}\right)[/latex]

15. [latex]\frac{1}{2}\left[\begin{array}{rrrr}\hfill 2& \hfill 1& \hfill -1& \hfill -1\\ \hfill 0& \hfill 1& \hfill 1& \hfill -1\\ \hfill 0& \hfill -1& \hfill 1& \hfill 1\\ \hfill 0& \hfill 1& \hfill -1& \hfill 1\end{array}\right][/latex]

16. [latex]\frac{1}{39}\left[\begin{array}{rrrr}\hfill 3& \hfill 2& \hfill 1& \hfill -7\\ \hfill 18& \hfill -53& \hfill 32& \hfill 10\\ \hfill 24& \hfill -36& \hfill 21& \hfill 9\\ \hfill -9& \hfill 46& \hfill -16& \hfill -5\end{array}\right][/latex]

17. 50% oranges, 25% bananas, 20% apples

18. 10 straw hats, 50 beanies, 40 cowboy hats

19. Tom ate 6, Joe ate 3, and Albert ate 3.

20. 124 oranges, 10 lemons, 8 pomegranates

Solving Systems with Cramer’s Rule

1. A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0.

2. The inverse does not exist.

3. [latex]-2[/latex]

4. [latex]7[/latex]

5. [latex]0[/latex]

6. [latex]3[/latex]

7. [latex]224[/latex]

8. [latex]-17.03[/latex]

9. [latex]\left(1,1\right)[/latex]

10. [latex]\left(\frac{1}{2},\frac{1}{3}\right)[/latex]

11. [latex]\left(15,12\right)[/latex]

12. [latex]\left(1,3,2\right)[/latex]

13. [latex]\left(-1,0,3\right)[/latex]

14. [latex]\left(\frac{1}{2},1,2\right)[/latex]

15. Infinite solutions

16. $7,000 in first account, $3,000 in second account.

17. 120 children, 1,080 adult

18. 4 gal yellow, 6 gal blue

19. 13 green tomatoes, 17 red tomatoes

20. Strawberries 18%, oranges 9%, kiwi 10%