If |adj(adj A)| = |A|⁹, then the order of the square matrix A 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 x^a y^b = e^m, x^c y^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

If A is a square matrix of order n, then |adj A| = ______________

If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______

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 A is a non-singular matrix then IA^-1| = ______

If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.

If A is a non-singular matrix of odd order, prove that |adj A| is positive

Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

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

Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)

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

Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

If A is a square matrix such that A³ = I, then prove that A is non-singular.

Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

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

Decrypt the received encoded message \(\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right] \) with the encryption matrix \(\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right] \) and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.

Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.

Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \) 

Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.

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 inverse (if it exists) of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is Rs. 150. The cost of the two dosai, two idlies and four vadais is Rs. 200. The cost of five dosai, four idlies and two vadais is Rs. 250. The family has Rs. 350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had ?

Investigate for what values of λ and μ the system of linear equations x  +  2y  +  z  =  7 ,   x  +  y  +  λz   =  μ ,   x  +  3y  −  5z   =  5 has
(i) no solution 
(ii) a unique solution 
(iii) an infinite number of solutions

If the system of equations px + by + cz = 0, ax + qy + cz = 0, ax + by + rz = 0 has a non-trivial solution and p ≠ a, q ≠ b, r ≠ c, prove that \(\frac { p }{ p-a } +\frac { q }{ q-b } +\frac { r }{ r-c } =2\).

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)

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

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.