{"id":328,"date":"2026-02-02T19:45:50","date_gmt":"2026-02-02T19:45:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=328"},"modified":"2026-02-02T20:27:24","modified_gmt":"2026-02-02T20:27:24","slug":"matrices-and-matrix-operations-get-stronger-answer-key-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/matrices-and-matrix-operations-get-stronger-answer-key-2\/","title":{"raw":"Matrices and Matrix Operations: Get Stronger Answer Key","rendered":"Matrices and Matrix Operations: Get Stronger Answer Key"},"content":{"raw":"

Matrices and Matrix Operations<\/h2>\r\n1.\u00a0No, 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.\r\n\r\n4. No. Matrices can only be multiplied if the inner dimensions are equal. For example, a 2x3<\/strong> matrix cannot be multiplied by another 2<\/strong>x3 matrix since 3 does not equal 2.\r\n\r\n7.\u00a0[latex]\\left[\\begin{array}{cc}11& 19\\\\ 15& 94\\\\ 17& 67\\end{array}\\right][\/latex]\r\n\r\n9.\u00a0[latex]\\left[\\begin{array}{cc}-4& 2\\\\ 8& 1\\end{array}\\right][\/latex]\r\n\r\n11.\u00a0Undidentified; dimensions do not match\r\n\r\n13.\u00a0[latex]\\left[\\begin{array}{cc}9& 27\\\\ 63& 36\\\\ 0& 192\\end{array}\\right][\/latex]\r\n\r\n15.\u00a0[latex]\\left[\\begin{array}{cccc}-64& -12& -28& -72\\\\ -360& -20& -12& -116\\end{array}\\right][\/latex]\r\n\r\n17.\u00a0[latex]\\left[\\begin{array}{ccc}1,800& 1,200& 1,300\\\\ 800& 1,400& 600\\\\ 700& 400& 2,100\\end{array}\\right][\/latex]\r\n\r\n19.\u00a0[latex]\\left[\\begin{array}{cc}20& 102\\\\ 28& 28\\end{array}\\right][\/latex]\r\n\r\n21.\u00a0[latex]\\left[\\begin{array}{ccc}60& 41& 2\\\\ -16& 120& -216\\end{array}\\right][\/latex]\r\n\r\n23.\u00a0[latex]\\left[\\begin{array}{ccc}-68& 24& 136\\\\ -54& -12& 64\\\\ -57& 30& 128\\end{array}\\right][\/latex]\r\n\r\n25.\u00a0Undefined; dimensions do not match.\r\n\r\n27.\u00a0[latex]\\left[\\begin{array}{ccc}-8& 41& -3\\\\ 40& -15& -14\\\\ 4& 27& 42\\end{array}\\right][\/latex]\r\n\r\n29.\u00a0[latex]\\left[\\begin{array}{ccc}-840& 650& -530\\\\ 330& 360& 250\\\\ -10& 900& 110\\end{array}\\right][\/latex]\r\n\r\n31.\u00a0[latex]\\left[\\begin{array}{cc}-350& 1,050\\\\ 350& 350\\end{array}\\right][\/latex]\r\n\r\n33.\u00a0Undefined; inner dimensions do not match.\r\n\r\n35.\u00a0[latex]\\left[\\begin{array}{cc}1,400& 700\\\\ -1,400& 700\\end{array}\\right][\/latex]\r\n\r\n37.\u00a0[latex]\\left[\\begin{array}{cc}332,500& 927,500\\\\ -227,500& 87,500\\end{array}\\right][\/latex]\r\n\r\n39.\u00a0[latex]\\left[\\begin{array}{cc}490,000& 0\\\\ 0& 490,000\\end{array}\\right][\/latex]\r\n\r\n41.\u00a0[latex]\\left[\\begin{array}{ccc}-2& 3& 4\\\\ -7& 9& -7\\end{array}\\right][\/latex]\r\n\r\n43.\u00a0[latex]\\left[\\begin{array}{ccc}-4& 29& 21\\\\ -27& -3& 1\\end{array}\\right][\/latex]\r\n\r\n45.\u00a0[latex]\\left[\\begin{array}{ccc}-3& -2& -2\\\\ -28& 59& 46\\\\ -4& 16& 7\\end{array}\\right][\/latex]\r\n\r\n47.\u00a0[latex]\\left[\\begin{array}{ccc}1& -18& -9\\\\ -198& 505& 369\\\\ -72& 126& 91\\end{array}\\right][\/latex]\r\n\r\n49.\u00a0[latex]\\left[\\begin{array}{cc}0& 1.6\\\\ 9& -1\\end{array}\\right][\/latex]\r\n\r\n51.\u00a0[latex]\\left[\\begin{array}{ccc}2& 24& -4.5\\\\ 12& 32& -9\\\\ -8& 64& 61\\end{array}\\right][\/latex]\r\n\r\n53.\u00a0[latex]\\left[\\begin{array}{ccc}0.5& 3& 0.5\\\\ 2& 1& 2\\\\ 10& 7& 10\\end{array}\\right][\/latex]\r\n

Solving Systems with Gaussian Elimination<\/h2>\r\n1.\u00a0Yes. 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.\r\n\r\n7.\u00a0[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]\r\n\r\n9.\u00a0[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]\r\n\r\n11.\u00a0[latex]\\begin{array}{l}-2x+5y=5\\\\ 6x - 18y=26\\end{array}[\/latex]\r\n\r\n13.\u00a0[latex]\\begin{array}{l}3x+2y=13\\\\ -x - 9y+4z=53\\\\ 8x+5y+7z=80\\end{array}[\/latex]\r\n\r\n15.\u00a0[latex]\\begin{array}{l}4x+5y - 2z=12\\hfill \\\\ \\text{ }y+58z=2\\hfill \\\\ 8x+7y - 3z=-5\\hfill \\end{array}[\/latex]\r\n\r\n17.\u00a0No solutions\r\n\r\n19.\u00a0[latex]\\left(-1,-2\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(6,7\\right)[\/latex]\r\n\r\n25.\u00a0[latex]\\left(\\frac{1}{5},\\frac{1}{2}\\right)[\/latex]\r\n\r\n27.\u00a0[latex]\\left(x,\\frac{4}{15}\\left(5x+1\\right)\\right)[\/latex]\r\n\r\n31.\u00a0[latex]\\left(\\frac{196}{39},-\\frac{5}{13}\\right)[\/latex]\r\n\r\n33.\u00a0[latex]\\left(31,-42,87\\right)[\/latex]\r\n\r\n35.\u00a0[latex]\\left(\\frac{21}{40},\\frac{1}{20},\\frac{9}{8}\\right)[\/latex]\r\n\r\n37.\u00a0[latex]\\left(\\frac{18}{13},\\frac{15}{13},-\\frac{15}{13}\\right)[\/latex]\r\n\r\n39.\u00a0[latex]\\left(x,y,\\frac{1}{2}\\left(1 - 2x - 3y\\right)\\right)[\/latex]\r\n\r\n43.\u00a0[latex]\\left(125,-25,0\\right)[\/latex]\r\n\r\n45.\u00a0[latex]\\left(8,1,-2\\right)[\/latex]\r\n\r\n53.\u00a0860 red velvet, 1,340 chocolate\r\n\r\n55.\u00a04% for account 1, 6% for account 2\r\n\r\n57.\u00a0$126\r\n\r\n59.\u00a0Banana was 3%, pumpkin was 7%, and rocky road was 2%\r\n\r\n61.\u00a0100 almonds, 200 cashews, 600 pistachios\r\n

Solving Systems with Inverses<\/h2>\r\n3.\u00a0No, 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.\r\n\r\n7.\u00a0[latex]AB=BA=\\left[\\begin{array}{cc}1& 0\\\\ 0& 1\\end{array}\\right]=I[\/latex]\r\n\r\n11.\u00a0[latex]AB=BA=\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right]=I[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{1}{29}\\left[\\begin{array}{cc}9& 2\\\\ -1& 3\\end{array}\\right][\/latex]\r\n\r\n17.\u00a0There is no inverse\r\n\r\n19.\u00a0[latex]\\frac{4}{7}\\left[\\begin{array}{cc}0.5& 1.5\\\\ 1& -0.5\\end{array}\\right][\/latex]\r\n\r\n21.\u00a0[latex]\\frac{1}{17}\\left[\\begin{array}{ccc}-5& 5& -3\\\\ 20& -3& 12\\\\ 1& -1& 4\\end{array}\\right][\/latex]\r\n\r\n25.\u00a0[latex]\\left[\\begin{array}{ccc}18& 60& -168\\\\ -56& -140& 448\\\\ 40& 80& -280\\end{array}\\right][\/latex]\r\n\r\n27.\u00a0[latex]\\left(-5,6\\right)[\/latex]\r\n\r\n31.\u00a0[latex]\\left(\\frac{1}{3},-\\frac{5}{2}\\right)[\/latex]\r\n\r\n37.\u00a0[latex]\\left(5,0,-1\\right)[\/latex]\r\n\r\n41.\u00a0[latex]\\frac{1}{690}\\left(65,-1136,-229\\right)[\/latex]\r\n\r\n43.\u00a0[latex]\\left(-\\frac{37}{30},\\frac{8}{15}\\right)[\/latex]\r\n\r\n45.\u00a0[latex]\\left(\\frac{10}{123},-1,\\frac{2}{5}\\right)[\/latex]\r\n\r\n47.\u00a0[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]\r\n\r\n49.\u00a0[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]\r\n\r\n55.\u00a050% oranges, 25% bananas, 20% apples\r\n\r\n57.\u00a010 straw hats, 50 beanies, 40 cowboy hats\r\n\r\n59.\u00a0Tom ate 6, Joe ate 3, and Albert ate 3.\r\n\r\n61.\u00a0124 oranges, 10 lemons, 8 pomegranates\r\n

Solving Systems with Cramer's Rule<\/h2>\r\n1.\u00a0A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product\u2014even if it does end up being 0.\r\n\r\n3.\u00a0The inverse does not exist.\r\n\r\n5.\u00a0[latex]-2[\/latex]\r\n\r\n7.\u00a0[latex]7[\/latex]\r\n\r\n11.\u00a0[latex]0[\/latex]\r\n\r\n15.\u00a0[latex]3[\/latex]\r\n\r\n19.\u00a0[latex]224[\/latex]\r\n\r\n23.\u00a0[latex]-17.03[\/latex]\r\n\r\n25.\u00a0[latex]\\left(1,1\\right)[\/latex]\r\n\r\n27.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{3}\\right)[\/latex]\r\n\r\n33.\u00a0[latex]\\left(15,12\\right)[\/latex]\r\n\r\n35.\u00a0[latex]\\left(1,3,2\\right)[\/latex]\r\n\r\n37.\u00a0[latex]\\left(-1,0,3\\right)[\/latex]\r\n\r\n39.\u00a0[latex]\\left(\\frac{1}{2},1,2\\right)[\/latex]\r\n\r\n43.\u00a0Infinite solutions\r\n\r\n53.\u00a0$7,000 in first account, $3,000 in second account.\r\n\r\n55.\u00a0120 children, 1,080 adult\r\n\r\n57.\u00a04 gal yellow, 6 gal blue\r\n\r\n59.\u00a013 green tomatoes, 17 red tomatoes\r\n\r\n61.\u00a0Strawberries 18%, oranges 9%, kiwi 10%\r\n\r\n63.\u00a0100 for movie 1, 230 for movie 2, 312 for movie 3","rendered":"

Matrices and Matrix Operations<\/h2>\n

1.\u00a0No, 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.<\/p>\n

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

7.\u00a0[latex]\\left[\\begin{array}{cc}11& 19\\\\ 15& 94\\\\ 17& 67\\end{array}\\right][\/latex]<\/p>\n

9.\u00a0[latex]\\left[\\begin{array}{cc}-4& 2\\\\ 8& 1\\end{array}\\right][\/latex]<\/p>\n

11.\u00a0Undidentified; dimensions do not match<\/p>\n

13.\u00a0[latex]\\left[\\begin{array}{cc}9& 27\\\\ 63& 36\\\\ 0& 192\\end{array}\\right][\/latex]<\/p>\n

15.\u00a0[latex]\\left[\\begin{array}{cccc}-64& -12& -28& -72\\\\ -360& -20& -12& -116\\end{array}\\right][\/latex]<\/p>\n

17.\u00a0[latex]\\left[\\begin{array}{ccc}1,800& 1,200& 1,300\\\\ 800& 1,400& 600\\\\ 700& 400& 2,100\\end{array}\\right][\/latex]<\/p>\n

19.\u00a0[latex]\\left[\\begin{array}{cc}20& 102\\\\ 28& 28\\end{array}\\right][\/latex]<\/p>\n

21.\u00a0[latex]\\left[\\begin{array}{ccc}60& 41& 2\\\\ -16& 120& -216\\end{array}\\right][\/latex]<\/p>\n

23.\u00a0[latex]\\left[\\begin{array}{ccc}-68& 24& 136\\\\ -54& -12& 64\\\\ -57& 30& 128\\end{array}\\right][\/latex]<\/p>\n

25.\u00a0Undefined; dimensions do not match.<\/p>\n

27.\u00a0[latex]\\left[\\begin{array}{ccc}-8& 41& -3\\\\ 40& -15& -14\\\\ 4& 27& 42\\end{array}\\right][\/latex]<\/p>\n

29.\u00a0[latex]\\left[\\begin{array}{ccc}-840& 650& -530\\\\ 330& 360& 250\\\\ -10& 900& 110\\end{array}\\right][\/latex]<\/p>\n

31.\u00a0[latex]\\left[\\begin{array}{cc}-350& 1,050\\\\ 350& 350\\end{array}\\right][\/latex]<\/p>\n

33.\u00a0Undefined; inner dimensions do not match.<\/p>\n

35.\u00a0[latex]\\left[\\begin{array}{cc}1,400& 700\\\\ -1,400& 700\\end{array}\\right][\/latex]<\/p>\n

37.\u00a0[latex]\\left[\\begin{array}{cc}332,500& 927,500\\\\ -227,500& 87,500\\end{array}\\right][\/latex]<\/p>\n

39.\u00a0[latex]\\left[\\begin{array}{cc}490,000& 0\\\\ 0& 490,000\\end{array}\\right][\/latex]<\/p>\n

41.\u00a0[latex]\\left[\\begin{array}{ccc}-2& 3& 4\\\\ -7& 9& -7\\end{array}\\right][\/latex]<\/p>\n

43.\u00a0[latex]\\left[\\begin{array}{ccc}-4& 29& 21\\\\ -27& -3& 1\\end{array}\\right][\/latex]<\/p>\n

45.\u00a0[latex]\\left[\\begin{array}{ccc}-3& -2& -2\\\\ -28& 59& 46\\\\ -4& 16& 7\\end{array}\\right][\/latex]<\/p>\n

47.\u00a0[latex]\\left[\\begin{array}{ccc}1& -18& -9\\\\ -198& 505& 369\\\\ -72& 126& 91\\end{array}\\right][\/latex]<\/p>\n

49.\u00a0[latex]\\left[\\begin{array}{cc}0& 1.6\\\\ 9& -1\\end{array}\\right][\/latex]<\/p>\n

51.\u00a0[latex]\\left[\\begin{array}{ccc}2& 24& -4.5\\\\ 12& 32& -9\\\\ -8& 64& 61\\end{array}\\right][\/latex]<\/p>\n

53.\u00a0[latex]\\left[\\begin{array}{ccc}0.5& 3& 0.5\\\\ 2& 1& 2\\\\ 10& 7& 10\\end{array}\\right][\/latex]<\/p>\n

Solving Systems with Gaussian Elimination<\/h2>\n

1.\u00a0Yes. 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.<\/p>\n

7.\u00a0[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]<\/p>\n

9.\u00a0[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]<\/p>\n

11.\u00a0[latex]\\begin{array}{l}-2x+5y=5\\\\ 6x - 18y=26\\end{array}[\/latex]<\/p>\n

13.\u00a0[latex]\\begin{array}{l}3x+2y=13\\\\ -x - 9y+4z=53\\\\ 8x+5y+7z=80\\end{array}[\/latex]<\/p>\n

15.\u00a0[latex]\\begin{array}{l}4x+5y - 2z=12\\hfill \\\\ \\text{ }y+58z=2\\hfill \\\\ 8x+7y - 3z=-5\\hfill \\end{array}[\/latex]<\/p>\n

17.\u00a0No solutions<\/p>\n

19.\u00a0[latex]\\left(-1,-2\\right)[\/latex]<\/p>\n

21.\u00a0[latex]\\left(6,7\\right)[\/latex]<\/p>\n

25.\u00a0[latex]\\left(\\frac{1}{5},\\frac{1}{2}\\right)[\/latex]<\/p>\n

27.\u00a0[latex]\\left(x,\\frac{4}{15}\\left(5x+1\\right)\\right)[\/latex]<\/p>\n

31.\u00a0[latex]\\left(\\frac{196}{39},-\\frac{5}{13}\\right)[\/latex]<\/p>\n

33.\u00a0[latex]\\left(31,-42,87\\right)[\/latex]<\/p>\n

35.\u00a0[latex]\\left(\\frac{21}{40},\\frac{1}{20},\\frac{9}{8}\\right)[\/latex]<\/p>\n

37.\u00a0[latex]\\left(\\frac{18}{13},\\frac{15}{13},-\\frac{15}{13}\\right)[\/latex]<\/p>\n

39.\u00a0[latex]\\left(x,y,\\frac{1}{2}\\left(1 - 2x - 3y\\right)\\right)[\/latex]<\/p>\n

43.\u00a0[latex]\\left(125,-25,0\\right)[\/latex]<\/p>\n

45.\u00a0[latex]\\left(8,1,-2\\right)[\/latex]<\/p>\n

53.\u00a0860 red velvet, 1,340 chocolate<\/p>\n

55.\u00a04% for account 1, 6% for account 2<\/p>\n

57.\u00a0$126<\/p>\n

59.\u00a0Banana was 3%, pumpkin was 7%, and rocky road was 2%<\/p>\n

61.\u00a0100 almonds, 200 cashews, 600 pistachios<\/p>\n

Solving Systems with Inverses<\/h2>\n

3.\u00a0No, 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.<\/p>\n

7.\u00a0[latex]AB=BA=\\left[\\begin{array}{cc}1& 0\\\\ 0& 1\\end{array}\\right]=I[\/latex]<\/p>\n

11.\u00a0[latex]AB=BA=\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right]=I[\/latex]<\/p>\n

13.\u00a0[latex]\\frac{1}{29}\\left[\\begin{array}{cc}9& 2\\\\ -1& 3\\end{array}\\right][\/latex]<\/p>\n

17.\u00a0There is no inverse<\/p>\n

19.\u00a0[latex]\\frac{4}{7}\\left[\\begin{array}{cc}0.5& 1.5\\\\ 1& -0.5\\end{array}\\right][\/latex]<\/p>\n

21.\u00a0[latex]\\frac{1}{17}\\left[\\begin{array}{ccc}-5& 5& -3\\\\ 20& -3& 12\\\\ 1& -1& 4\\end{array}\\right][\/latex]<\/p>\n

25.\u00a0[latex]\\left[\\begin{array}{ccc}18& 60& -168\\\\ -56& -140& 448\\\\ 40& 80& -280\\end{array}\\right][\/latex]<\/p>\n

27.\u00a0[latex]\\left(-5,6\\right)[\/latex]<\/p>\n

31.\u00a0[latex]\\left(\\frac{1}{3},-\\frac{5}{2}\\right)[\/latex]<\/p>\n

37.\u00a0[latex]\\left(5,0,-1\\right)[\/latex]<\/p>\n

41.\u00a0[latex]\\frac{1}{690}\\left(65,-1136,-229\\right)[\/latex]<\/p>\n

43.\u00a0[latex]\\left(-\\frac{37}{30},\\frac{8}{15}\\right)[\/latex]<\/p>\n

45.\u00a0[latex]\\left(\\frac{10}{123},-1,\\frac{2}{5}\\right)[\/latex]<\/p>\n

47.\u00a0[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]<\/p>\n

49.\u00a0[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]<\/p>\n

55.\u00a050% oranges, 25% bananas, 20% apples<\/p>\n

57.\u00a010 straw hats, 50 beanies, 40 cowboy hats<\/p>\n

59.\u00a0Tom ate 6, Joe ate 3, and Albert ate 3.<\/p>\n

61.\u00a0124 oranges, 10 lemons, 8 pomegranates<\/p>\n

Solving Systems with Cramer’s Rule<\/h2>\n

1.\u00a0A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product\u2014even if it does end up being 0.<\/p>\n

3.\u00a0The inverse does not exist.<\/p>\n

5.\u00a0[latex]-2[\/latex]<\/p>\n

7.\u00a0[latex]7[\/latex]<\/p>\n

11.\u00a0[latex]0[\/latex]<\/p>\n

15.\u00a0[latex]3[\/latex]<\/p>\n

19.\u00a0[latex]224[\/latex]<\/p>\n

23.\u00a0[latex]-17.03[\/latex]<\/p>\n

25.\u00a0[latex]\\left(1,1\\right)[\/latex]<\/p>\n

27.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{3}\\right)[\/latex]<\/p>\n

33.\u00a0[latex]\\left(15,12\\right)[\/latex]<\/p>\n

35.\u00a0[latex]\\left(1,3,2\\right)[\/latex]<\/p>\n

37.\u00a0[latex]\\left(-1,0,3\\right)[\/latex]<\/p>\n

39.\u00a0[latex]\\left(\\frac{1}{2},1,2\\right)[\/latex]<\/p>\n

43.\u00a0Infinite solutions<\/p>\n

53.\u00a0$7,000 in first account, $3,000 in second account.<\/p>\n

55.\u00a0120 children, 1,080 adult<\/p>\n

57.\u00a04 gal yellow, 6 gal blue<\/p>\n

59.\u00a013 green tomatoes, 17 red tomatoes<\/p>\n

61.\u00a0Strawberries 18%, oranges 9%, kiwi 10%<\/p>\n

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