Solutions 31: Matrices and Matrix Operations

Solutions to Try Its

1. 
A+B=[2116   03]+[314253]=[2+31+11+(4)6+(2)0+53+3]=[523450]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=[8264]-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. 
[641228723602012116]\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. 
[6041216120216]\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right]


23. 
[68241365412645730128]\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right]


25. Undefined; dimensions do not match.

27. 
[841340151442742]\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right]


29. 
[84065053033036025010900110]\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. 
[1,4007001,400700]\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right]


37. 
[332,500927,500227,50087,500]\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. 
[234797]\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right]


43. 
[429212731]\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right]


45. 
[3222859464167]\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right]


47. 
[11891985053697212691]\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right]


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


51. 
[2244.51232986461]\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. 
[100010001]even[100001010]odd\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \text{even}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \text{odd}


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