Any set $$\{ {v_1},{v_2},...,{v_p}\} \in {R^n}$$ is linearly dependent if

• A) p<n
• B) p>n
• C)
• D)

$${\text{Set}} \left\{ {\left( {\begin{array}{*{20}{c}} 1 \\\ 2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 0 \\\ 0 \end{array}} \right)} \right\} {\text{is Linearly - - - - - in}}~ {\mathbb{R}^2}.$$

• A) Independent
• B) Dependent
• C)
• D)

$$\eqalign{ & If A = {A^t} ( Where A is a square matrix), then which of the following \cr & is the most appropriate option for A \cr}$$

• A) A is a scalar matrix.
• B) A is invertible matrix.
• C) A is singular matrix.
• D) A is symetric matrix.

A set of two vectors $$\{ {v_1},{v_2}\}$$ is linearly dependent if and only if one of the vector is

• A) Additive inverse of the other
• B) Multiple of the other
• C)
• D)

The columns of a matrix A are ------------------- if and only if the equation $Ax=0$ has only the trivial solution.

• A) linearly independent
• B) linearly dependent
• C)
• D)

Which of the following is true for the partitioned matrices $Y = [\,Q\,\,\,\,\,\,\,\,R\,]\,\,\,{\rm{and}}\,\,\,Z = [\,U\,\,\,\,\,\,\,\,V\,]\,,$ where sub-matrices $Q$ and $R$ have the same sizes as $U$ and $V$ respectively?

• A) $$Y + Z = [\,Q + U\,\,\,\,\,\,\,R + V\,]\,$$
• B) $$Y + Z = [\,QU\,\,\,\,\,\,\,RV\,]\,$$
• C) $$Y + Z = \left[ \begin{array}{l} \,QU\\\RV \end{array} \right]$$
• D) $$Y + Z = \left[ \begin{array}{l} \,Q + U\\\R + V\, \end{array} \right]$$

Which of the following is true for the speed of convergence of both the Jacobi and Gauss-Seidel sequences?

• A) It depends on how much the diagonal entries dominate the corresponding row sums.
• B) It depends on how much the diagonal entries dominate the corresponding column sums.
• C) It depends on how much the diagonal entries dominate the preceding column sums.
• D) It depends on how much the diagonal entries dominate the preceding row sums.

The Jacobi’s iteration converges if a square matrix A is diagonally dominant.

• A) False
• B) True
• C)
• D)

If $X = \left[ \begin{array}{l} \,M\\\N\, \end{array} \right]\,\,{\rm{and}}\,\,Y = \left[ {\,Q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,} \right]$ ( where $M$ , $N$ , $Q$ and $P$ are square sub-matrices of same size ) , then which of the following is possible ?

• A) The product $XY$ and $YX$ both are defined.
• B) The product $XY$ and $YX$ both are not defined.
• C) The product $XY$ is not defined but $YX$ is defined.
• D) The product $XY$ is defined but $YX$ is not defined.

As soon as an approximate solution of a linear system is sufficiently accurate for practical work then which of the following is true for an iterative process?

• A) The process may be stopped.
• B) The reduced row-echelon method may be used.
• C) The row-echelon method may be used.
• D) The process may be carried on.

In LU-decomposition, if A is the square matrix of order m, then the order of both lower and upper triangular matrices will be ______.

• A) $m+1$
• B) $m$
• C) $2m$
• D) $m^2$

If a system of equations is solved using the Gauss-Seidel method, then which of the following is the most appropriate answer about the matrix M that is derived from the coefficient matrix ?

• A) All of its entries below and above the diagonal must be zero.
• B) All of its entries above the diagonal must be zero.
• C) All of its entries below the diagonal must be zero.
• D) All of its entries on the diagonal must be zero.

The Gauss-Seidel method is applicable to strictly _________ dominant matrix.

• A) column wise
• B) row wise
• C) diagonally
• D) both row and column wise

A matrix in which each descending diagonal from left to right is constant is called a _____ matrix.

• A) diagonal
• B) diagonal-constant
• C) block upper triangular
• D) block lower triangular

The Jacobi’s method is a method of solving a matrix equation on a matrix that has _______ zeros along its main diagonal.

• A) No
• B) Two
• C) more than two
• D) more than two

In Jacobi’s method, the rate of convergence is quite ---------- compared with other methods.

• A) None of these
• B) Slow
• C) Fast
• D) Speedy

The Jacobi’s method is a method of solving a matrix equation on a matrix that has two zeros along its main diagonal.

• A) True
• B) False
• C)
• D)

A partitioned matrix ‘A’ is said to be block diagonal if the matrices on the main diagonal are square and all other position matrices are ______.

• A) Unit
• B) Nonzero skew symmetric
• C) Nonzero symmetric
• D) Zero

LU-decomposition converts the problem of solving the single system A x = b into the problem of solving the two systems, L y = b and U x = y, these systems are easy to solve because their co-efficient matrices are_______.

• A) Identity
• B) Diagonal
• C) Identity
• D) Triangular

If a square matrix A can be reduced to row echelon form with _____ interchanges, then A has an LU-decomposition.

• A) No row
• B) Row
• C) Column
• D) No column