1 : A set of orthogonal vectors - 1) are linearly independent 2) can be converted to orthonormal vectors by dividing them by their respective length 3) adhere to trace(transpose(A).B) = 0 4) are linearly independent and the corollary is ALSO true Which of the above statements is TRUE -
Options:
  1. 1,2,3
  2. 1,3,4
  3. 1,3
  4. 2,4

Answer: Option 1

Explanation:

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2 : Rank -
Options:
  1. of a null matrix doesn't exists
  2. of any matrix other than null matrix is > or = 1
  3. of any matrix other than null matrix is > or = 2
  4. of any matrix other than null matrix is > or = 0

Answer: Option 2

Explanation:

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3 : Rank of a matrix -
Options:
  1. can be calculated ONLY if the matrix is square
  2. is the order of largest square sub-matrix present in a matrix
  3. is the same as no. of linearly dependent row or coloumn vectors in the matrix
  4. is the order of largest non-singular square sub-matrix present in a matrix

Answer: Option 4

Explanation:

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4 : Let A be a square matrix, then, the solution of characteristic equation |A-kI|=0 is equal to -
Options:
  1. Product of Eigen values of A
  2. Eigen values of A-kI
  3. Product of Eigen values of A-kI
  4. None of the above

Answer: Option 1

Explanation:

1) |A| is itself a solution of this characteristic equation which otherwise gives eigen values of A 2) Product of eigen values of A = |A| Discuss this in Forums

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5 : In an under-determined (means No. of equations < No. of variables) system of Linear non-homogeneous eq's -
Options:
  1. We can have only consistent and infinite solution
  2. We can have either inconsistent or consistent and infinite solution
  3. We can have a consistent and unique solution
  4. We can have every possible solution

Answer: Option 2

Explanation:

In an under-determined system,either - 1) Rank(A) is not equal to Rank(A|B) OR 2) If Rank(A) is equal to Rank(A|B), Rank(A) will always be less than n So, we will ONLY get infinite no. of solutions. Discuss this in Forums

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6 : The sum of all the diagonal elements of a matrix A -
Options:
  1. Is equal to the product of eigen values
  2. is always less than |A|
  3. is always greater than |A|
  4. is equal to the trace of A

Answer: Option 4

Explanation:

By definition, Trace of a square matrix = Sum of its diagonal elements = Sum of all its eigen values Discuss this in Forums

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7 : If a row or coloumn of a matrix A undergoes transformation given by Ri+kRj or Ci+kCj respectively, the value of determinant of new matrix is-
Options:
  1. k times |A|
  2. zero
  3. remains equal to |A|
  4. changes to a new value not related to |A|

Answer: Option 3

Explanation:

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8 : Let A,B,C,D,E be n*n matrices, each with non-zero determinant and ABCDE = I, where I = Identity matrix of order n, Then C =
Options:
  1. (B^-1)*(A^-1)*(E^-1)*(D*-1)
  2. (A^-1)*(B^-1)*(D^-1)*(E*-1)
  3. (ABDE)^-1
  4. (EDBA)*-1

Answer: Option 1

Explanation:

ABCDE = I BCDE = (A^-1) Pre-Multiplying with A^-1 on both sides CDE = (B^-1)*(A^-1) Pre-Multiplying with B^-1 on both sides CD = (B^-1)*(A^-1)*(E^-1) Post-Multiplying with E^-1 on both sides C = (B^-1)*(A^-1)*(E^-1)*(D^-1) Post-Multiplying with D^-1 on both sides Discuss this in Forums

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9 : A = a1 a2 a3 b1 b2 b3 c1 c2 c3 Then, a1*cof(b1) + a2*cof(b2) + a3*cof(b3) =
Options:
  1. |A|
  2. Has no meaning
  3. Zero
  4. None of the above

Answer: Option 3

Explanation:

The sum of the products of the elements of any row (or coloumn) with cofactors of some other row (or coloumn) respectively is ALWAYS zero. Discuss this in Forums

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10 : Let A = 1 1 3 5 2 6 -2 -1 -3 It is seen that A^3 = 0. So, A is a-
Options:
  1. Singular Matrix
  2. Nilpotent Matrix of order 3
  3. Null Matrix
  4. None of the above

Answer: Option 2

Explanation:

If for any matrix A, A^n = 0, then A is called a Nilpotent Matrix of order n Discuss this in Forums

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Fairy tales are more than true: not because they tell us that dragons exist, but because they tell us that dragons can be beaten.
-Neil Gaiman Coraline