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.

4. No. Matrices can only be multiplied if the inner dimensions are equal. For example, a 2x3 matrix cannot be multiplied by another 2x3 matrix since 3 does not equal 2.

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

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

11. Undidentified; dimensions do not match

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

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

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

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

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

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

25. Undefined; dimensions do not match.

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

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

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

33. Undefined; inner dimensions do not match.

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

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

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

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

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

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

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

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

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

53. [latex]\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\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.

7. [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]

9. [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]

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

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

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

17. No solutions

19. [latex]\left(-1,-2\right)[/latex]

21. [latex]\left(6,7\right)[/latex]

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

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

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

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

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

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

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

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

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

53. 860 red velvet, 1,340 chocolate

55. 4% for account 1, 6% for account 2

57. $126

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

61. 100 almonds, 200 cashews, 600 pistachios

Solving Systems with Inverses

3. 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.

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

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

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

17. There is no inverse

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

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

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

27. [latex]\left(-5,6\right)[/latex]

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

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

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

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

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

47. [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]

49. [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]

55. 50% oranges, 25% bananas, 20% apples

57. 10 straw hats, 50 beanies, 40 cowboy hats

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

61. 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.

3. The inverse does not exist.

5. [latex]-2[/latex]

7. [latex]7[/latex]

11. [latex]0[/latex]

15. [latex]3[/latex]

19. [latex]224[/latex]

23. [latex]-17.03[/latex]

25. [latex]\left(1,1\right)[/latex]

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

33. [latex]\left(15,12\right)[/latex]

35. [latex]\left(1,3,2\right)[/latex]

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

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

43. Infinite solutions

53. $7,000 in first account, $3,000 in second account.

55. 120 children, 1,080 adult

57. 4 gal yellow, 6 gal blue

59. 13 green tomatoes, 17 red tomatoes

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

63. 100 for movie 1, 230 for movie 2, 312 for movie 3