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

(a)

0

(b)

2

(c)

3

(d)

1

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

(a)

n −1

(b)

n

(c)

n +1

(d)

n²

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)

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

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

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

Rank of a null matrix is _______.

(a)

0

(b)

-1

(c)

\(\infty \)

(d)

1

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 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

For any 2 x 2 matrix, if A (adj A = \(\left| \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right| \) then IAI is _______

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10

If A is a square matrix of order n, then |Adj A|=______

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|A|^n-1

If A is a square matrix such that A² = I, then A^-1=_______

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A

The number of solutions of the system of equations 2x + Y - z = 7, x - 3y + 2z = 1,x + 4y - 3z = 5 is______

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0

A set of values of the variable x₁,x₂,...x_n satisfying all the equations simultaneously is called__________ of the system

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Solution

Rank of a matrix

(1)

\(\ge 0\)

Rank of a zero matrix is

(2)

1

If A is of rank 2, then adj A is of rank

(3)

0

For the system of equations AX = B, the solution is X = A^-1 B provided

(4)

\(|A|\neq 0\)

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

(5)

inconsistent