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

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

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

Rank of a null matrix is _______.

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

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

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

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

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

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

The system of non-homogeneous equations will have.
(a) unique solution
(b) Infinitely many solutions
(c) No solution
(d) Trivial solution

Rank of a 2 x 2 matrix may be
(a) 0
(b)1
(c) 2
(d)3

The transition probabilities P_jk satisfy
(a) \({ P }_{ \mu }>0\)
(b) \(\sum _{ k }^{ 1 }{ { P }_{ jk } } =1\)
(c) \({ P }_{ jk }\le 0\)
(d) \({ P }_{ jk }>1\)

Which one is incorrect?
(a) \({ R }_{ 1 }\rightarrow { R }_{ 1 }+{ R }_{ 2 }\)
(b) \({ C }_{ 1 }\rightarrow { C }_{ 1 }-{ 2C }_{ 2 }\)
(c) \({ R }_{ 3 }\leftrightarrow { R }_{ 1 }\)
(d) \({ R }_{ 1 }\leftrightarrow { C }_{ 1 }\)

If IAI = 0, then
(a) A is a singular matrix
(b) System has either no solution or infinitely many solutions
(c) No solution
(d) non-singular matrix

If \(A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix} \right] \) then the value of |adj A| is ___________

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

The system of equation x + y + z = 2, 3x - y + 2z = 6 and 3x + y - z = -18 has________solution

The system of linear equations x + y + Z = 2, 2x + Y - z = 3, 3x + 2y + kz = 4 has a unique solution if k is_______

The value on for which the system of equations x + y + z = 5, x + 2y + 3z = 9, \(x+3y+\lambda z=\mu \) is __________

Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them by rank method.

For what values of the parameter λ, will the following equations fail to have unique solution: 3x − y+λz = 1, 2x + y + z = 2, x + 2y − λz = −1 by rank method.

Show that the equations x + y + z = 6, x + 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

For what value of x, the matrix
\(A=\left| \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right| \) is singular?

If \(\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right) \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) =\left( \begin{matrix} 1 \\ -1 \\ 0 \end{matrix} \right) \) find x, y and z

Find the rank of the matrix \(\begin{pmatrix} -5 & -7 \\ 5 & 7 \end{pmatrix}\)

Find the rank of the matrix \(\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right) \)

Find the rank of the matrix \(A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right) \)

Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

Solve: 2x + 3y = 5, 6x + 5y = 11

The following table represents the number of shares of two companies A and B during the month of January and February and it also gives the amount in rupees invested by Ravi during these two months for the purchase of shares of two companies. Find the the price per share of A and B purchased during both the months

The total cost of 11 pencils and 3 erasers is Rs. 64 and the total cost of 8 pencils and 3 erasers is Rs. 49. Find the cost of each pencil and each eraser by Cramer’s rule.

The price of 3 Business Mathematics books, 2 Accountancy books and one Commerce book is Rs. 840. The price of 2 Business Mathematics books, one Accountancy book and one Commerce book is Rs. 570. The price of one Business Mathematics book, one Accountancy book and 2 Commerce books is Rs. 630. Find the cost of each book by using Cramer’s rule.

A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R? (Cramer's rule).