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Determinants

Short Answer Type

31. A matrix A, of order 3 × 3, has determinant 4. Find the value of | 3 A |

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Multiple Choice Questions

32. Which of the following is correct

(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(C) Determinant is a number associated to a square matrix.

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Short Answer Type

33.

Solve space for space straight x space colon open square brackets table row cell 2 straight x end cell cell space 0 end cell cell space space 1 end cell row 0 cell space 1 end cell cell space space 2 end cell row cell negative 1 end cell cell space 2 end cell cell space space 0 end cell end table close square brackets space equals space open vertical bar table row 1 cell space space space space 0 end cell cell space space space 0 end cell row 2 cell space space space space 3 end cell cell space minus 4 end cell row 3 cell negative 3 end cell cell space space 5 end cell end table close vertical bar

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34.

If space open square brackets table row 1 cell space space space 1 end cell cell space space minus 2 end cell row 2 cell space space 1 end cell cell space space minus 3 end cell row 5 cell space space 4 end cell cell space space minus 9 end cell end table close square brackets space space Find space open vertical bar space straight A space close vertical bar

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35.

If space space straight A equals open square brackets table row cell 1 space space end cell cell space space 0 end cell cell space space space space space 1 end cell row 0 cell space space 1 end cell cell space space space space space 2 end cell row 0 cell space space 0 end cell cell space space space space 4 end cell end table close square brackets

then show that | 3A | = 27 | A |.

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36.

Syntax error from line 1 column 244 to line 1 column 251.

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Multiple Choice Questions

37.

Let space space space space space straight A equals space open square brackets table row 1 cell sin space straight theta end cell 1 row cell negative sin space straight theta end cell 1 cell sin space straight theta end cell row cell negative 1 end cell cell negative sin space straight theta end cell 1 end table close square brackets space comma space w h e r e space 0 space element of space ө space 2 space pi. space T h e n

(A) Det(A) = 0
(B) Det(A) ∈ (2, ∞)
(C) Det(A) ∈ (2, 4)
(C) Det(A) ∈ (2, 4)

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Short Answer Type

38. The value of the determinant remains unchanged if its rows and columns are interchanged. $Verify space the space property space for space space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar$

space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar Expanding space the space determinant space along space first space row comma space we space have increment space space equals 2 space open vertical bar table row 0 cell space space space space space space 4 end cell row 5 cell space space minus 7 end cell end table close vertical bar minus left parenthesis negative 3 right parenthesis space open vertical bar table row 6 cell space space space space 4 end cell row 1 cell space space space minus 7 end cell end table close vertical bar space plus 5 space open vertical bar table row 6 cell space space 0 end cell row 1 cell space space 5 end cell end table close vertical bar space space space space space equals space 2 left parenthesis 0 minus 20 right parenthesis plus 3 left parenthesis negative 42 minus 4 right parenthesis plus 5 left parenthesis 30 minus 0 right parenthesis equals negative 40 minus 138 plus 150 equals negative 28 By space interchanging space rows space and space columns comma space we space get increment subscript 1 space equals space space space space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar space equals 2 space space open vertical bar table row 0 cell space space space space space space 4 end cell row 5 cell space space minus 7 end cell end table close vertical bar minus left parenthesis negative 3 right parenthesis space open vertical bar table row 6 cell space space space space 4 end cell row 1 cell space space space minus 7 end cell end table close vertical bar space plus 5 space open vertical bar table row 6 cell space space 0 end cell row 1 cell space space 5 end cell end table close vertical bar space space space space space equals 2 space left parenthesis 0 minus 20 right parenthesis plus 3 left parenthesis negative 42 minus 4 right parenthesis plus 5 left parenthesis 30 minus 0 right parenthesis space space space space space equals negative 40 minus 138 plus 150 equals negative 28 therefore space space space increment space equals space increment subscript 1 space Hence space property space is space verified.

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39. If any two rows of a determinant are interchanged, then sign of determinant

verify space this space property space for space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space space space space space 5 end cell row 6 cell space space space space space 0 end cell cell space space space space space space 4 end cell row 1 cell space space space space 5 end cell cell space space minus 7 end cell end table close vertical bar

changes.

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40. Without expanding, prove that

open vertical bar table row 9 cell space space space space space space 9 end cell cell space space 12 end cell row 1 cell space space minus 3 end cell cell space space minus 4 end cell row 1 cell space space space space space space 9 end cell cell space 12 end cell end table close vertical bar space equals 0

70 Views