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Guides d'étude > Precalculus II

Solutions for Matrices and Matrix Operations

Solutions to Try Its

1. A+B=[2116   0−3]+[31−4−253]=[2+31+11+(−4)6+(−2)0+5−3+3]=[52−3450]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] 2. −2B=[−8−2−6−4]-2B=\left[\begin{array}{cc}-8& -2\\ -6& -4\end{array}\right]

Solutions to Odd-Numbered Exercises

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 2×22\times 2 matrix and the second is a 2×32\times 3 matrix. [1234]+[654321]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right] has no sum. 3. Yes, if the dimensions of AA are m×nm\times n and the dimensions of BB are n×m,n\times m,\text{} both products will be defined. 5. Not necessarily. To find AB,AB,\text{} we multiply the first row of AA by the first column of BB to get the first entry of ABAB. To find BA,BA,\text{} we multiply the first row of BB by the first column of AA to get the first entry of BABA. Thus, if those are unequal, then the matrix multiplication does not commute. 7. [111915941767]\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right] 9. [−4281]\left[\begin{array}{cc}-4& 2\\ 8& 1\end{array}\right] 11. Undidentified; dimensions do not match 13. [92763360192]\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right] 15. [−64−12−28−72−360−20−12−116]\left[\begin{array}{cccc}-64& -12& -28& -72\\ -360& -20& -12& -116\end{array}\right] 17. [1,8001,2001,3008001,4006007004002,100]\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right] 19. [201022828]\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right] 21. [60412−16120−216]\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right] 23. [−6824136−54−1264−5730128]\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right] 25. Undefined; dimensions do not match. 27. [−841−340−15−1442742]\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right] 29. [−840650−530330360250−10900110]\left[\begin{array}{ccc}-840& 650& -530\\ 330& 360& 250\\ -10& 900& 110\end{array}\right] 31. [−3501,050350350]\left[\begin{array}{cc}-350& 1,050\\ 350& 350\end{array}\right] 33. Undefined; inner dimensions do not match. 35. [1,400700−1,400700]\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right] 37. [332,500927,500−227,50087,500]\left[\begin{array}{cc}332,500& 927,500\\ -227,500& 87,500\end{array}\right] 39. [490,00000490,000]\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right] 41. [−234−79−7]\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right] 43. [−42921−27−31]\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right] 45. [−3−2−2−285946−4167]\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right] 47. [1−18−9−198505369−7212691]\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right] 49. [01.69−1]\left[\begin{array}{cc}0& 1.6\\ 9& -1\end{array}\right] 51. [224−4.51232−9−86461]\left[\begin{array}{ccc}2& 24& -4.5\\ 12& 32& -9\\ -8& 64& 61\end{array}\right] 53. [0.530.521210710]\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right] 55. [100010001]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] 57. [100010001]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] 59. {B}^{n}=\left\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\text{ }n\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\text{ }n\text{odd}\text{.}\end{array}

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.