$$\begin{gathered} {\text{If the equation:}}\left( {\begin{array}{*{20}{c}} { - 2}&3 \\ 5&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{b_1}} \\ {{b_2}} \end{array}} \right){\text{has the solution for all }} {b_1},{b_2} \in \mathbb{R}, \ {\text{then}} \left( {\begin{array}{*{20}{c}} { - 2} \\ 5 \end{array}} \right) {\text{and}} \left( {\begin{array}{*{20}{c}} 3 \\ 1 \end{array}} \right) {\text{will span}} - - - - - . \\ \end{gathered} $$

• A) $$\mathbb{R}~space$$
• B) $${\text{Nothing}}$$
• C) $${\mathbb{R}^3}space$$
• D) $${\mathbb{R}^2}space $$

If the order of matrices A, B and C are 2*3, 3*15 and 15*100 respectively, then the order of Product ABC= -------.

• A) 2*100
• B) 200
• C) 405000
• D) 100*2

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

The matrix M = \left[ {\,\begin{array}{*{20}{c}} a&4&5&6\\\7&a&4&5\\\8&7&a&4\\\9&8&7&a \end{array}\,} \right] is a ______ matrix.

• A) Diagonal
• B) Toeplitz
• C) Identity
• D) Rectangular

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

Consider an LU-factorization, the matrix L is _____ and is called a unit lower triangular matrix.

• A) Singular
• B) Invertible
• C) zero
• D) Identity

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

{\text{Let A be the matrix of order 2}} \times {\text{3 and B be the matrix of order 3}} \times {\text{5,then which of the following is the order of the matrix AB?}}

• A) 2 \times 3
• B) 3 \times 3
• C) 2 \times 5
• D) 3 \times 5

An m x m ______ matrix is only square matrix that is both unit lower triangular and upper triangular.

• A) Scalar
• B) Diagonal
• C) Null
• D) Identity

Which of the following is the most suitable option while doing LU-factorization?

• A) Entries cannot be interchanged.
• B) Columns can be interchanged.
• C) Rows can be interchanged.
• D) Rows cannot be interchanged.

The matrix $ M = \left[ {\,\begin{array}{*{20}{c}} a&4&5&6\\\7&a&4&5\\\8&7&a&4\\\9&8&7&a \end{array}\,} \right]$ is a ______ matrix.

• A) Toeplitz
• B) Rectangular
• C) Diagonal
• D) Identity

If M=[3] then which of the following is the determinant of the matrix M?

• A) [3]
• B) 1
• C) 3
• D) [1]

If |D|=ad-bc is a determinant of an n x n matrix D then at what condition matrix D is invertible?

• A) When |D|=ad-bc=0
• B) When |D|=ad-bc is not equal to zero
• C) When |D|=ad-bc=|2|
• D) When |D|=ad-bc= [0 2]

If a multiple of one row of a square matrix A is added to another row to produce a matrix B, then which of the following condition is true?

• A) det(B). det(A) = det(A)
• B) det(A). det(B) = 0
• C) det(B) = k det(A)
• D) det(B) = det(A)

If a matrix A is invertible than adj(A) is also invertible.

• A) FALSE
• B) TRUE
• C)
• D)

If the absolute value of each diagonal entry exceeds the sum of the absolute values of the other entries in the same row then a matrix A is called:

• A) diagonally
• B) scalar
• C) invertible
• D) strictly diagonally dominant

According to determinant properties, the determinants of resulting matrix equal to k delta if elements of rows are-------

• A) Subtracting to constant k
• B) Multiplied to constant k
• C) Divided to constant k
• D) Added to constant k

At what condition the Cramer’s formula is valid for linear systems?

• A) When det(A) is equal to zero
• B) When matrix is m x n
• C) When matrix is n x n
• D) When det(A) in not equal to zero

If all the elements of one row is ‘0’ in a matrix A then which of the following about the determinant of the matrix is true?

• A) det (A)is not equal to'1'
• B) det (A)is not equal to '0'
• C) det (A)=1
• D) det (A)=0

Why inverse of the matrix A= [1 2] is NOT possible?

• A) Because it is an identity matrix.
• B) Because it is a square matrix.
• C) Because it is a rectangular matrix
• D) Because it is a zero matrix.