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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)

space space space space space straight A equals space open square brackets table row 1 cell space space space sin space straight theta end cell 1 row cell negative sin space straight theta end cell 1 cell space space sin space straight theta end cell row cell negative 1 end cell cell space minus sin space straight theta end cell 1 end table close square brackets open vertical bar space straight A space close vertical bar space equals space open vertical bar table row 1 cell space sin space straight theta end cell cell space space 1 end cell row cell negative sin space straight theta end cell cell space 1 end cell cell space space sin space straight theta end cell row cell negative 1 end cell cell space space minus sin space straight theta end cell 1 end table close vertical bar space space space space space space space space equals space space 1 cross times space open vertical bar table row 1 cell space space sin space straight theta end cell row cell negative sin space straight theta end cell 1 end table close vertical bar minus sin space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space space sin space straight theta end cell row cell negative 1 end cell 1 end table close vertical bar plus 1 space cross times space open vertical bar table row cell negative sin space straight theta end cell 1 row cell negative 1 end cell cell sin space straight theta end cell end table close vertical bar space space space space space space space space equals space space space 1 cross times left parenthesis 1 plus sin space straight theta squared space straight theta right parenthesis minus sin space straight theta space left parenthesis negative sin space straight theta space space plus sin space straight theta right parenthesis plus 1 cross times left parenthesis sin squared space straight theta plus 1 right parenthesis space space space space space space space equals space left parenthesis 1 plus sin squared space straight theta right parenthesis minus 0 plus left parenthesis sin squared space straight theta plus 1 right parenthesis equals 2 plus 2 space sin squared space straight theta space equals 2 space left parenthesis 1 plus sin squared straight theta right parenthesis we space know space that space space space space space space space space space space minus 1 space less or equal than space sin space straight theta space less or equal than 1 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 space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta therefore space space space space space space space space space space space 0 space less or equal than space sin space to the power of 2 space end exponent less or equal than 1 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 space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space 1 plus 0 space less or equal than space 1 plus sin space squared space straight theta less or equal than space 1 plus 1 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 space space space space for space all space straight theta space space space space space space space space space space space rightwards double arrow space space space space space space 1 plus 0 space less or equal than space 1 plus sin space squared space straight theta less or equal than 2 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 space space space space space space space space space space space for space all space straight theta space space space space space space space space rightwards double arrow space space space space space space 2 less or equal than 2 space left parenthesis 1 plus sin space squared space straight theta right parenthesis less or equal than 4 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 space space space space space space space space space space space space space for space all space straight theta space space space space space space space space rightwards double arrow space space space space space space 2 space less or equal than space space space open vertical bar space straight A space close vertical bar space less or equal than 4 therefore space left parenthesis straight D right parenthesis space is space correct space answer. space space space space space space space space

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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.

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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

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

Short Answer Type