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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

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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.

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 increment space equals space 2 space open vertical bar table row 0 cell space space space space 4 end cell row 5 cell space space minus 7 end cell end table close vertical bar space minus left parenthesis negative 3 right parenthesis space open square brackets table row 6 cell space space space space space 4 end cell row 1 cell space space minus 7 end cell end table close square brackets space plus 5 space open square brackets table row 6 cell space space 0 end cell row 1 cell space space 5 end cell end table close square brackets space space space space space equals space 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 equals negative 40 minus 138 plus 150 equals negative 28 Interchanging space rows space straight R 2 space and space straight R 3 space straight i. straight e. space straight R 2 space left right arrow space straight R 3 comma space we space have space increment subscript 1 space space end subscript equals space open vertical bar table row 2 cell space space space minus 3 end cell cell space space space space space space 5 end cell row 1 cell space space space space space space 5 end cell cell space space space space space minus 7 end cell row 6 cell space space space space space 0 end cell cell space space space space space space 4 end cell end table close vertical bar increment subscript 1 space space end subscript equals space space 2 space open vertical bar table row 5 cell space space space minus 7 end cell row 0 cell space space space space space space 4 end cell end table close vertical bar plus 3 space open square brackets table row 1 cell space space space space minus 7 end cell row 6 cell space space space space space space space space space 4 end cell end table close square brackets space plus 5 space open square brackets table row 1 cell space space space 5 end cell row 6 cell space space 0 end cell end table close square brackets space space space space space space space equals space space 2 left parenthesis 20 minus 0 right parenthesis plus 3 left parenthesis 4 plus 42 right parenthesis plus 5 left parenthesis 0 minus 30 right parenthesis equals 40 plus 138 minus 150 equals 28 therefore space space space increment subscript 1 space space end subscript equals space minus increment therefore space space property space is space verified

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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

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

Multiple Choice Questions

Short Answer Type

Multiple Choice Questions

Short Answer Type

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