For what value of k, the matrix \(A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right) \) has no inverse?

(a)

\(\frac { 3 }{ 10 } \)

(b)

\(\frac { 10 }{ 3 } \)

(c)

3

(d)

10

The rank of an n x n matrix each of whose elements is 2 is __________

(a)

1

(b)

2

(c)

n

(d)

n²

The value of \(\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right| \)

(a)

5²

(b)

0

(c)

5¹³

(d)

5⁹

If \(\left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| =\left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right| \) then x =

(a)

3

(b)

± 3

(c)

± 6

(d)

6

If A is a singular matrix, then Adj A is ___________

(a)

non-singular

(b)

singular

(c)

symmetric

(d)

not defined

If A, B are two n x n non-singular matrices, then ___________

(a)

AB is non-singular

(b)

AB is singular

(c)

(AB)^-1 = A^-1 B^-1

(d)

(AB)^-1 does not exit

The rank of the matrix \(\left(\begin{array}{cc} 2 & -4 \\ -1 & 2 \end{array}\right) \text { is }\)___________

(a)

1

(b)

2

(c)

0

(d)

8

The rank of the matrix \(\left(\begin{array}{cc} 7 & -1 \\ 2 & 1 \end{array}\right) \text { is }\)___________

(a)

9

(b)

2

(c)

1

(d)

5

Which of the following is not elementary transformation?

(a)

\(R_{i} \leftrightarrow R_{j}\)

(b)

\(R_{i} \rightarrow 2 R_{i}+R_{j}\)

(c)

\(C_{i} \rightarrow C_{j}+C_{i}\)

(d)

\(R_{i} \rightarrow R_{i}+C_{i}\)

Equivalent matrices are obtained by ___________

(a)

Taking Inverses

(b)

Taking transposes

(c)

Taking adjoints

(d)

Taking finite number of elementary transformation

In echelon form, which of the following is incorrect?

(a)

Every row of A which has all its entries O occurs below every row which had a non - zero entry.

(b)

The first non -zero entry in each non-zero row is 1

(c)

The number of zeros before the first non zero element in a row is less than the number of such zeros in the next row.

(d)

2 rows can have the same number of zeros before the first non-zero entry

If \(\Delta \neq 0\) then the system is ___________

(a)

consistent and has unique solution

(b)

consistent and has infinitely many solution

(c)

inconsistent

(d)

either consistent or inconsistent

\(\text { The rank of the matrix }\left(\begin{array}{rrr} 1 & -1 & 2 \\ 2 & -2 & 4 \\ 4 & -4 & 8 \end{array}\right) \text { is }\) ___________

(a)

1

(b)

2

(c)

3

(d)

4

The rank of the diagonal matrix \(\left(\begin{array}{ccccc} -1 & & & & \\ & 2 & & & \\ & & 0 & & \\ & & & -4 & \\ & & & & 0 \end{array}\right) \text { is }\) ___________

(a)

0

(b)

2

(c)

3

(d)

5

If A = (2 01) then rank of AA^T is ___________

(a)

1

(b)

2

(c)

3

(d)

0