$$\begin{gathered} {x_1} + 2{x_2} + 3{x_3} = 7 \hfill \\\\ 4{x_1} + {x_2} + 2{x_3} = 2 \hfill \\\\ - 4{x_1} + 3{x_2} + 9{x_3} = 4 \hfill \\\\\\ \end{gathered}$$ The augmented matrix for the system is -----------

• A) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}} \right]$
• B) $$\left( {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}} \right)$$
• C) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$
• D) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$

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

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

$Let{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T:{R^2} \to {R^2}{\kern 1pt} {\kern 1pt} {\kern 1pt} be{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} transformation{\kern 1pt} {\kern 1pt} {\kern 1pt} T({x_1},{x_2}) = (x,0).{\kern 1pt} {\kern 1pt} {\kern 1pt} The{\kern 1pt} {\kern 1pt} {\kern 1pt} null{\kern 1pt} {\kern 1pt} {\kern 1pt} space{\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} \ker nel)\,N(T)\,\,of\,T\,\,is$

• A) $(0,1)$
• B) $(0, x_2)$
• C) $(1, 0)$
• D) $(x_1,0)$

Which one of the following is a diagonal matrix?

• A) $$\left( {\begin{array}{*{20}{c}} 2&2&0 \\\\\\ 0&2&0 \\\\\\ 0&0&1 \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} 2&0&0 \\\\\\ 0&1&0 \\\\\\ 0&0&1 \end{array}} \right)$$
• C)
• D)

We can add the matrices of ______________.

• A) different order
• B) same number of columns
• C) same number of rows.
• D) same order

$${\text{In }}{\mathbb{R}^2},{\text{Linearly dependent vectors lie on the same}} - - - - - .$$

• A) Line
• B) Plane
• C)
• D)

The determinant of a triangular matrix is the sum of the entries of the main diagonal.

• A) False
• B) True
• C)
• D)

A transformation $T:{R^n} \to {R^m}$ Is a rule that assigns to each vector x in ${R^n}$ , an image vector T(x) in ${R^m}$. The set ${R^n}$ is called the__________ , and ${R^m}$ is called the _______.

• A) domain of T, co-domain of T
• B) co-domain of T, domain of T
• C)
• D)

Which one of the following is a null matrix?

• A) $$\left( {\begin{array}{*{20}{c}} 0&0&0 \\\\\\\\\ 0&0&0 \\\\\\\\\ 0&0&0 \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} 1&0&0 \\\\\\\\\ 2&0&0 \\\\\\\\\ 0&0&1 \end{array}} \right)$$
• C)
• D)

A null space is a vector space.

• A) False
• B) True
• C)
• D)

$${\text{Vectors}} \left( {\begin{array}{*{20}{c}} { - 2} \\\ 3 \end{array}} \right){\text{ and}} \left( {\begin{array}{*{20}{c}} { - 66} \\\ {99} \end{array}} \right){\text{ are linearly }} - - - - - .$$

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

$${\text{If }}A = \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&6 \end{array}} \right)\mathop \sim \limits^{{\text{Row Equivalent}}} \left( {\begin{array}{*{20}{c}} 1&2 \\\ 0&0 \end{array}} \right), {\text{then the Columns of}} ~{\text{A}} ~{\text{are}} ~{\text{Linearly}} {\text{ - - - - - - - }}$$

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

$${\text{Vector}} \left( {\begin{array}{*{20}{c}} { - 1} \\\ 0 \end{array}} \right) {\text{is Linearly}} - - - - - - .$$

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

The two vectors $(-2,1)$ and $(1,2)$ are

• A) Perpendicular to each other
• B) Pointing in the opposite direction of each other
• C) Linearly dependent of each other
• D) None of the above

What is the maximum possible number of pivots in a 4x6 matrix ?

• A) 6
• B) 8
• C) 4
• D) 10

If A= transpose of A( where A is a square matrix ), then which of the following is the most appropriate option for A?

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

A system of linear equations is said to be homogeneous if it can be written in the form ______________.

• A) AX = 0
• B) AX = B
• C)
• D)

$${\text{Vectors}} \left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right) {\text{and}} \left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right) {\text{are linearly}} - - - - - .$$

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

Let A be the matrix of order 2X3 and B be the matrix of order 3X5,then which of the following is the order of the matrix AB?

• A) 2x5
• B) 3X5
• C) 3X3
• D) 2X3

$${\text{Since vector}}\left( {\begin{array}{*{20}{c}} 2 \\\ 3 \end{array}} \right) {\text{lies in the span}} \left\{ {\left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right)} \right\} {\text{then the vectors }}\left( {\begin{array}{*{20}{c}} 2 \\\ 3 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right) {\text{and}} \left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right) {\text{are Linearly }} - - - -$$

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