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Matrices

Short Answer Type

61. Find.v and y, if

2 space open square brackets table row 1 3 row 0 straight x end table close square brackets plus open square brackets table row straight y 0 row 1 2 end table close square brackets equals open square brackets table row 5 6 row 1 8 end table close square brackets

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62. Solve the equation for x, y, z and t, if

2 space open square brackets table row x z row y t end table close square brackets plus 3 open square brackets table row 1 cell negative 1 end cell row 0 2 end table close square brackets equals 3 open square brackets table row 3 5 row 4 6 end table close square brackets

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63.

If space straight x open parentheses table row 2 row 3 end table close parentheses plus straight y open square brackets table row cell negative 1 end cell row 1 end table close square brackets equals open square brackets table row 10 row 5 end table close square brackets space

find the values of x and y.

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64. Given

3 open square brackets table row straight x straight y row straight z straight w end table close square brackets space equals space open square brackets table row straight x 6 row cell negative 1 end cell cell 2 space straight w end cell end table close square brackets plus open square brackets table row 4 cell straight x plus straight y end cell row cell straight z plus straight w end cell 3 end table close square brackets

, find the values of .x, y, z and w.

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65. Solve the matrix equation

space space space space space space open square brackets table row cell straight x squared end cell row cell straight y squared end cell end table close square brackets minus 3 space open square brackets table row x row cell 2 space y end cell end table close square brackets equals open square brackets table row cell negative 2 end cell row 9 end table close square brackets space

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Long Answer Type

66. 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.

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Multiple Choice Questions

67.

X, Z are matrices of order 2 x 2 X p respectively.
If n = p, then the order of the matrix 7X – 5Z is :

p x 2
2 x n
n x 3
n x 3

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Short Answer Type

68.

Let A = $open square brackets table row 2 4 row 3 2 end table close square brackets comma space straight B equals open square brackets table row 1 3 row cell negative 2 end cell 5 end table close square brackets$

Find each of the following : (i) AB (ii) BA

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69.

Find AB, if A = $open square brackets table row 6 9 row 2 3 end table close square brackets space and space space straight B space equals space open square brackets table row 2 6 0 row 7 9 8 end table close square brackets$

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70. If A=

open square brackets table row 1 cell negative 2 end cell 3 row cell negative 4 end cell 2 5 end table close square brackets space and space straight B space equals space open square brackets table row 1 3 row 4 5 row 2 1 end table close square brackets

then find AB, BA.

Show that AB ≠ BA.

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