If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is

If A, B and C are invertible matrices of some order, then which one of the following is not true?

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

If \(\rho\) (A) = \(\rho\)([A| B]), then the system AX = B of linear equations is

If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

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

Every homogeneous system ______

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

Cramer's rule is applicable only when ______

In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________

In the system of liner equations with 3 unknowns If \(\rho\) (A) = \(\rho\) ([A|B]) =1, the system has ________

If A = [2 0 1] then the rank of AA^T is ______

If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.

Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

Find the rank of the following matrices by minor method or show that the rank of matrix is 3
\(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right] \)

Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).

Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

Solve the following systems of linear equations by Cramer’s rule:
5x − 2y +16 = 0, x + 3y − 7 = 0

In a competitive examination, one mark is awarded for every correct answer while \(\frac { 1 }{ 4 }\) mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly ? (Use Cramer’s rule to solve the problem).

Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method :
2x  −  y  =  8 ,   3x  +  2y  =  −2.

Solve the following systems of linear equations by Cramer’s rule:
 \(\frac { 3 }{ x } \) + 2y = 12, \(\frac { 2 }{ x } \) + 3y = 13

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).

Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4

Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2

The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.