If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is

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

The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

If A is a non-singular matrix of odd order them
1) Order of A is 2m + 1
(2) Order of A is 2m + 2
(3) |adj A| is positive
(4) IAI ≠ 0

A matrix which is obtained from an identity matrix by applying only one elementary transformation is
(1) Identity matrix
(2) Elementary matrix
(3) Square matrix
(4) Equivalent to identify matrix

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

Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 3 & 2 & 5 \\ 1 & 1 & 2 \\ 3 & 3 & 6 \end{matrix} \right] \) 

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

Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 6 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 0 \\ \begin{matrix} 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -9 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)

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

For the matrix A, if A³ = I, then find A^-1.

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

Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).

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.

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer’s rule to solve the problem).

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

The prices of three commodities A, B and C are Rs. x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B. Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn Rs. 15,000, Rs. 1,000 and Rs. 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)