If A = (1 2 3), then the rank of AA^T is ________.

(a)

0

(b)

2

(c)

3

(d)

1

The rank of m x n matrix whose elements are unity is ________.

(a)

0

(b)

1

(c)

m

(d)

n

if T = \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right) \) is a transition probability matrix, then at equilibrium A is equal to ________.

(a)

\(\frac { 1 }{ 4 } \)

(b)

\(\frac { 1 }{ 5 } \)

(c)

\(\frac { 1 }{ 6 } \)

(d)

\(\frac { 1 }{ 8 } \)

If A = \(\begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}\),then \(\rho (A)\) is ________.

(a)

0

(b)

1

(c)

2

(d)

n

The rank of the matrix  \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is ________.

(a)

0

(b)

1

(c)

2

(d)

3

The rank of the unit matrix of order n is ________.

(a)

n −1

(b)

n

(c)

n +1

(d)

n²

If \(\rho (A)\) = r  then which of the following is correct?

(a)

all the minors of order r which does not vanish

(b)

A has at least one minor of order r which does not vanish

(c)

A has at least one (r+1) order minor which vanishes

(d)

all (r+1) and higher order minors should not vanish

If A =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AA^T is ________.

(a)

0

(b)

1

(c)

2

(d)

3

If the rank of the matrix  \(\left( \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right) \)  is 2. Then \(\lambda \) is ________.

(a)

1

(b)

2

(c)

3

(d)

only real number

The rank of the diagonal matrix\(\left( \begin{matrix} 1 & & \\ & 2 & \\ & & -3 \end{matrix}\\ \quad \quad \quad \quad \quad \quad \quad \begin{matrix} 0 & & \\ & 0 & \\ & & 0 \end{matrix} \right) \)

(a)

0

(b)

2

(c)

3

(d)

5

If  \(T=\begin{array}{l} A \\ B \end{array}\left(\begin{array}{ll} 0.7 & 0.3 \\ 0.6 & x \end{array}\right)\) is a transition probability matrix, then the value of x is ________.

(a)

0.2

(b)

0.3

(c)

0.4

(d)

0.7

Which of the following is not an elementary transformation?

(a)

\({ R }_{ i }\leftrightarrow { R }_{ j }\)

(b)

\({ R }_{ i }\rightarrow { 2R }_{ i }+{ 2C }_{ j }\)

(c)

\({ R }_{ i }\rightarrow { 2R }_{ i }-{ 4R }_{ j}\)

(d)

\({ C }_{ i }\rightarrow { C }_{ i }+{ 5C }_{ j }\)

If \(\rho (A)=\rho (A,B)\) then the system is ________.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

Consistent

(d)

inconsistent

If \(\rho (A)=\rho (A,B)\) the number of unknowns, then the system is______.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

consistent

(d)

inconsistent

If \(\rho(A) \neq \rho(A, B)\), then the system is _______.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

inconsistent

(d)

consistent

In a transition probability matrix, all the entries are greater than or equal to _______.

(a)

2

(b)

1

(c)

0

(d)

3

If the number of variables in a non-homogeneous system AX = B is n, then the system possesses a unique solution only when _______.

(a)

\(\rho (A)=\rho (A,B)>n\)

(b)

\(\rho(A)=\rho(A, B)=n\)

(c)

\(\rho (A)=\rho (A,B) < n\)

(d)

none of these

The system of equations 4x + 6y = 5, 6x + 9y = 7 has _______.

(a)

a unique solution

(b)

no solution

(c)

infinitely many solutions

(d)

none of these

For the system of equations x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 _______.

(a)

there is only one solution

(b)

there exists infinitely many solutions

(c)

there is no solution

(d)

None of these

If \(\left| A \right| \neq 0,\) then A is _______.

(a)

non-singular matrix

(b)

singular matrix

(c)

zero matrix

(d)

none of these

The system of linear equations x + y + z = 2, 2x + y − z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to _______.

(a)

4

(b)

0

(c)

-4

(d)

1

Cramer’s rule is applicable only to get an unique solution when _______.

(a)

\({ \triangle }_{ z }\neq 0\)

(b)

\({ \triangle }_{ x }\neq 0\)

(c)

\({ \triangle } \neq 0\)

(d)

\({ \triangle }_{ y }\neq 0\)

If \(\frac { { a }_{ 1 } }{ x } +\frac { { b }_{ 1 } }{ y } ={ c }_{ 1 },\frac { { a }_{ 2 } }{ x } +\frac { { b }_{ 2 } }{ y } ={ c }_{ 2 },\) \({ \triangle }_{ 1= }\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } \end{vmatrix}, \ { \triangle }_{ 2 }=\begin{vmatrix} { b }_{ 1 } & { c }_{ 1 } \\ { b }_{ 2 } & { c }_{ 2 } \end{vmatrix}{ \triangle }_{ 3 }=\begin{vmatrix} { c }_{ 1 } & { a }_{ 1 } \\ { c }_{ 2 } & a_{ 2 } \end{vmatrix}\) then (x, y) is _______.

(a)

\(\left( \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } ,\frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } } \right) \)

(b)

\(\left( \frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } }, \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } \right) \)

(c)

\(\left( \frac { { \triangle }_{ 1 } }{ { \triangle }_{ 2 } } ,\frac { { \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right) \)

(d)

\(\left( \frac { { -\triangle }_{ 1 } }{ { \triangle }_{ 2 } }, \frac { {- \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right) \)

\(\left| { A }_{ n\times n } \right| \) = 3 \(\left| adjA \right| \) = 243 then the value n is _______.

(a)

4

(b)

5

(c)

6

(d)

7

Rank of a null matrix is _______.

(a)

0

(b)

-1

(c)

\(\infty \)

(d)

1