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     If space straight A equals space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space space is space such space that space straight A squared space equals space straight I comma space then left parenthesis straight A right parenthesis space straight l space plus space straight alpha squared space plus space βγ space equals space 0 space left parenthesis straight B right parenthesis space straight l – straight alpha squared plus space βγ space equals space 0 space space left parenthesis straight C right parenthesis space 1 space – straight alpha squared space – space βγ space equals space 0 space left parenthesis straight D right parenthesis space straight l space plus straight alpha squared – space βγ space equals space 0

    space straight A equals space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets now space straight A squared space equals 1 space rightwards double arrow space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space open square brackets table row straight alpha cell space space space space space space straight beta end cell row straight gamma cell space space minus straight alpha end cell end table close square brackets space equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space open square brackets table row cell straight alpha to the power of 2 space space end exponent plus space βγ end cell cell γβ minus αβ end cell row cell γα minus γα end cell cell βγ plus straight alpha squared end cell end table close square brackets equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow open square brackets table row cell straight alpha to the power of 2 space space end exponent plus space βγ end cell 0 row 0 cell straight alpha to the power of 2 space space end exponent plus space βγ end cell end table close square brackets space equals space open square brackets table row 1 cell space space space 1 end cell row 0 cell space space space 0 end cell end table close square brackets rightwards double arrow space by space definition space of space equality space of space matrices comma straight alpha to the power of 2 space space end exponent plus space βγ space equals 1 space space or space 1 minus straight alpha to the power of 2 space space end exponent plus space βγ space equals 0 therefore space space left parenthesis straight C right parenthesis space is space correct space answer. space
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    Answer
    Two farmers Ramkishan and Gurcharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.
    September Sales (in Rupees)
    space space space space space space italic space italic space italic space B a s m a t i italic space italic space italic space italic space italic space P e r m a l italic space italic space italic space italic space italic space N a u r a A equals open square brackets table row cell 10 comma 000 space space space space space space space space 20 comma 000 space space space space space 30 comma 000 end cell row cell 50 comma 000 space space space space space space space space 30 comma 000 space space space space space 10 comma 000 space end cell end table close square brackets space table row Ramkrishan row cell Gurcharan space Singh end cell end table
    October Sales (in Rupees)
    space space space space space space italic space italic space italic space Basmati italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space Naura straight A equals open square brackets table row cell 5 comma 000 space space space space space space space space 20 comma 000 space space space space space 6 comma 000 end cell row cell 20 comma 000 space space space space space space space space 10 comma 000 space space space space space 10 comma 000 space end cell end table close square brackets space table row Ramkrishan row cell Gurcharan space Singh end cell end table

    (i) Find the combined sales in September and October for each farmer in each variety.
    (ii) Find the decrease in sales from September to October.
    (iii) If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for cach variety sold in October.
    The sale (in Rupees) of these varieties of rice by both the fanners in the month of September and October are given by the following matrices A and B.

    September Sales (in Rupees)

    space space space space space space italic space italic space italic space Basmati italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space Naura straight A equals open square brackets table row cell 10 comma 000 space space space space space space space space 20 comma 000 space space space space space 30 comma 000 end cell row cell 50 comma 000 space space space space space space space space 30 comma 000 space space space space space 10 comma 000 space end cell end table close square brackets space table row Ramkrishan row cell Gurcharan space Singh end cell end table

    October Sales (in Rupees)

    space space space space space space italic space italic space italic space Basmati italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space Naura straight A equals open square brackets table row cell 5 comma 000 space space space space space space space space 20 comma 000 space space space space space 6 comma 000 end cell row cell 20 comma 000 space space space space space space space space 10 comma 000 space space space space space 10 comma 000 space end cell end table close square brackets space table row Ramkrishan row cell Gurcharan space Singh end cell end table

    (i) Combined sales in September and October for each farmer in each variety is given by

    space space space space space space italic space italic space italic space Basmati italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space Naura straight A equals open square brackets table row cell 10 comma 000 plus 5 comma 000 space space space space space space space space space space space space space 20 comma 000 plus 10 comma 000 space space space space space 30 comma 000 plus 6 comma 000 end cell row cell 50 comma 000 plus 20 comma 000 space space space space space space space space space 30 comma 000 plus 10 comma 000 space space space space space 10 comma 000 plus 10 comma 000 space end cell end table close square brackets space space equals space open square brackets table row cell 15 comma 000 space space space 30 comma 000 space space space 36 comma 000 end cell row cell 70 comma 000 space space space space 40 comma 000 space space space space space 20 comma 000 space space space end cell end table close square brackets space table row Ramkishan row cell Gurcharan space Singh end cell end table

    (ii) Change in sales from September to October is given by

    space space space space space space italic space italic space italic space Basmati italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space Naura straight A equals open square brackets table row cell 10 comma 000 minus 5 comma 000 space space space space space space space space space space space space space 20 comma 000 minus 10 comma 000 space space space space space 30 comma 000 minus 6 comma 000 end cell row cell 50 comma 000 minus 20 comma 000 space space space space space space space space space 30 comma 000 minus 10 comma 000 space space space space space 10 comma 000 minus 10 comma 000 space end cell end table close square brackets space space equals space open square brackets table row cell 5 comma 000 space space space 10 comma 000 space space space 24 comma 000 end cell row cell 30 comma 000 space space space space 20 comma 000 space space space space space 0 space space space end cell end table close square brackets space table row Ramkishan row cell Gurcharan space Singh end cell end table

    left parenthesis iii right parenthesis space 2 percent sign space of space straight B equals 2 over 100 cross times straight B equals 0.02 cross times straight B space space space space space space italic space italic space italic space italic space italic space italic space Basmati italic space italic space italic space italic space italic space Permal italic space italic space italic space italic space italic space italic space italic space italic space italic space Naura equals 0.02 space open square brackets table row cell 5 comma 000 end cell cell space space space space space space 10 comma 000 end cell cell space space space space space 6 comma 000 end cell row cell 20 comma 000 end cell cell space space space space space space space 10 comma 000 end cell cell space space space space space 10 comma 000 end cell end table close square brackets space space equals space open square brackets table row 100 200 120 row 400 200 200 end table close square brackets space table row Ramkishan row cell space space space Gurcharan space Singh end cell end table

    Thus, in October Ramkishan receives Rs. 100, Rs. 200 and Rs. 120 as profit in the sale of each variety of rice, respectively, and Gurcharan Singh receives profit of Rs. 400, Rs. 200 and Rs. 200 in the sale of each variety of rice, respectively.
    102 Views
    Answer
    If space straight A space is space square space matrix space such space that space straight A squared equals space straight A comma space then space left parenthesis straight I space plus space straight A right parenthesis cubed space – space 7 straight A space is space equal space to space left parenthesis straight A right parenthesis space straight A space comma space left parenthesis straight B right parenthesis space straight I – space straight A space left parenthesis straight C right parenthesis space 1 space left parenthesis straight D right parenthesis space 3 straight A
    We space have space straight A squared space equals straight A space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis therefore space space straight A cubed space equals space space straight A squared space. straight A space equals straight A space. straight A space equals straight A to the power of 2 space end exponent space equals space straight A space space... left parenthesis 2 right parenthesis now space space left parenthesis space straight I plus straight A right parenthesis space minus 7 straight A space equals space left parenthesis straight I plus 3 straight A plus 3 straight A to the power of 2 space end exponent space plus space space straight A cubed space right parenthesis space minus 7 straight A space space space space open square brackets table row cell because space of space left parenthesis 1 right parenthesis space end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight I plus 3 space straight A plus 3 space straight A space plus to the power of space space space straight A space right parenthesis space minus 7 straight A space space space space space open square brackets table row cell because space of space left parenthesis 1 right parenthesis space. space left parenthesis 2 right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight I therefore space left parenthesis straight C right parenthesis space space is space correct space answer. space space space space space space space space space space space space space space space
    75 Views
    Answer
    Let space straight A equals space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space space and space straight I comma space the space identity space matrix space of space order space 2. show space that space straight I plus straight A equals space left parenthesis straight I plus straight A right parenthesis space open square brackets table row cell cos space straight alpha end cell cell space space minus sin space straight alpha end cell row cell sin space straight alpha end cell cell space space space space cos space straight alpha end cell end table close square brackets
    Here space straight A equals space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space comma space straight I space equals space open square brackets table row 1 cell space space 0 end cell row 0 cell space space 1 end cell end table close square brackets straight L. straight H. straight S equals space straight I plus straight A space space space open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space 1 end cell end table close square brackets space space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space equals space space open square brackets table row 1 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 1 end cell end table close square brackets straight R. straight H. straight S equals space left parenthesis straight I minus straight A right parenthesis space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space open parentheses space open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space 1 end cell end table close square brackets minus space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space close parentheses space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space space space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space open square brackets table row cell cos space straight alpha space plus sin space straight alpha space tan begin inline style straight a over 2 end style space space space end cell cell negative sin space straight alpha space plus space cos space straight alpha space space tan begin inline style straight a over 2 end style end cell row cell negative cosα space tan begin inline style straight a over 2 space plus space sin space straight alpha end style end cell cell sin space straight alpha space tan straight alpha over 2 plus space cos space straight alpha end cell end table close square brackets equals space open square brackets table row cell 1 minus 2 space sin squared space begin inline style straight alpha over 2 end style space plus space 2 space sin squared space space begin inline style straight alpha over 2 end style space end cell cell space space space space space space minus tan space begin inline style straight alpha over 2 space open parentheses table row cell 2 cos squared space space space minus straight alpha over 2 end cell end table cosα space close parentheses end style end cell row cell tan space begin inline style straight alpha over 2 end style open parentheses table row cell 2 cos squared space begin inline style straight alpha over 2 end style space space minus end cell end table cosα space close parentheses end cell cell space 1 minus 2 space sin squared space begin inline style straight alpha over 2 end style space plus space 2 space sin squared space space begin inline style straight alpha over 2 end style end cell end table space space space space close square brackets equals space space open square brackets table row 1 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 1 end cell end table close square brackets therefore straight L. straight H. straight S equals straight R. straight H. straight S
    104 Views
    Answer
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     Assume Y, Z, W and P are the matrices of order 3 x k x 2 x p, n x 3 and p x k, respectively.
    The restrictions on n, k and p so that PY + WY will be defined are :
    (A) k = 3, p = n    (B) A:is arbitrary,p = 2
    (C) p is arbitrary, k = 3    (D)k = 2, p = 3

    Y, Z, W, P are of type 3 x k, 2 x p, n x 3, p x k respectively.
    Since PY is defined
    ∴number of columns of P = number of rows of Y
    ∴ k = 3
    Now PY is of type p x k and WY is of type n x k
    Since PY + WY is defined
    ∴ p = n
    ∴ (A) is correct answer.

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