{"id":1794,"date":"2015-11-12T18:30:44","date_gmt":"2015-11-12T18:30:44","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1794"},"modified":"2017-04-03T21:44:17","modified_gmt":"2017-04-03T21:44:17","slug":"solutions-23","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solutions-23\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\r\n1.\u00a0[latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex]\r\n\r\n2.\u00a0[latex]-2B=\\left[\\begin{array}{cc}-8& -2\\\\ -6& -4\\end{array}\\right][\/latex]\r\n

Solutions to Odd-Numbered Exercises<\/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\n3.\u00a0Yes, if the dimensions of [latex]A[\/latex] are [latex]m\\times n[\/latex] and the dimensions of [latex]B[\/latex] are [latex]n\\times m,\\text{}[\/latex] both products will be defined.\r\n\r\n5.\u00a0Not necessarily. To find [latex]AB,\\text{}[\/latex] we multiply the first row of [latex]A[\/latex] by the first column of [latex]B[\/latex] to get the first entry of [latex]AB[\/latex]. To find [latex]BA,\\text{}[\/latex] we multiply the first row of [latex]B[\/latex] by the first column of [latex]A[\/latex] to get the first entry of [latex]BA[\/latex]. Thus, if those are unequal, then the matrix multiplication does not commute.\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\r\n55.\u00a0[latex]\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]\r\n\r\n57.\u00a0[latex]\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]\r\n\r\n59.\u00a0[latex]{B}^{n}=[\/latex]\r\n\r\n[latex]{B}^{n=\\text{ even}}\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]\r\n\r\n[latex]{B}^{n=\\text{ odd}}\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 0& 1\\\\ 0& 1& 0\\end{array}\\right][\/latex]","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0[latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex]<\/p>\n

2.\u00a0[latex]-2B=\\left[\\begin{array}{cc}-8& -2\\\\ -6& -4\\end{array}\\right][\/latex]<\/p>\n

Solutions to Odd-Numbered Exercises<\/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

3.\u00a0Yes, if the dimensions of [latex]A[\/latex] are [latex]m\\times n[\/latex] and the dimensions of [latex]B[\/latex] are [latex]n\\times m,\\text{}[\/latex] both products will be defined.<\/p>\n

5.\u00a0Not necessarily. To find [latex]AB,\\text{}[\/latex] we multiply the first row of [latex]A[\/latex] by the first column of [latex]B[\/latex] to get the first entry of [latex]AB[\/latex]. To find [latex]BA,\\text{}[\/latex] we multiply the first row of [latex]B[\/latex] by the first column of [latex]A[\/latex] to get the first entry of [latex]BA[\/latex]. Thus, if those are unequal, then the matrix multiplication does not commute.<\/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

55.\u00a0[latex]\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]<\/p>\n

57.\u00a0[latex]\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]<\/p>\n

59.\u00a0[latex]{B}^{n}=[\/latex]<\/p>\n

[latex]{B}^{n=\\text{ even}}\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]<\/p>\n

[latex]{B}^{n=\\text{ odd}}\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 0& 1\\\\ 0& 1& 0\\end{array}\\right][\/latex]<\/p>\n\n\t\t\t

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