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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.

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

Here $A equals open square brackets table row bold 6 bold 9 row bold 2 bold 3 end table close square brackets bold space bold B bold space bold equals bold space open square brackets table row bold 2 bold 6 bold 0 row bold 7 bold 9 bold 8 end table close square brackets$

The matrix A is of type 2 x 2 and B is of type 2 x 3.
Number of columns of A = number of rows of B = 2. So AB is defined

$AB equals open square brackets table row 6 9 row 2 3 end table close square brackets open square brackets table row 2 6 0 row 7 9 8 end table close square brackets space space space space equals space open square brackets table row cell left parenthesis 6 right parenthesis left parenthesis 2 right parenthesis left parenthesis 9 right parenthesis left parenthesis 7 right parenthesis end cell cell left parenthesis 6 right parenthesis left parenthesis 6 right parenthesis plus left parenthesis 9 right parenthesis left parenthesis 9 right parenthesis end cell cell left parenthesis 6 right parenthesis left parenthesis 0 right parenthesis plus left parenthesis 9 right parenthesis left parenthesis 8 right parenthesis end cell row cell left parenthesis 2 right parenthesis left parenthesis 2 right parenthesis plus left parenthesis 3 right parenthesis left parenthesis 7 right parenthesis end cell cell left parenthesis 2 right parenthesis left parenthesis 6 right parenthesis plus left parenthesis 3 right parenthesis left parenthesis 9 right parenthesis end cell cell left parenthesis 2 right parenthesis left parenthesis 0 right parenthesis plus left parenthesis 3 right parenthesis left parenthesis 8 right parenthesis end cell end table close square brackets space space space space equals space open square brackets table row cell 12 plus 63 end cell cell 36 plus 81 end cell cell 0 plus 72 end cell row cell 4 plus 21 end cell cell 12 plus 27 end cell cell 0 plus 24 end cell end table close square brackets therefore space space AB equals open square brackets table row 75 117 72 row 25 39 24 end table close square brackets$

Note 1. Here AB is defined. But BA is not defined as number of columns of B is not equal to number of rows of A.

Note 2. If A and B are square matrices of the same order, then AB and BA are both defined.

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