Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|² is equal to :

The ordered pair (a, b), for which the system of linear equations

3x $$-$$ 2y + z = b

5x $$-$$ 8y + 9z = 3

2x + y + az = $$-$$1

has no solution, is :

The system of equations

$$ - kx + 3y - 14z = 25$$

$$ - 15x + 4y - kz = 3$$

$$ - 4x + y + 3z = 4$$

is consistent for all k in the set

Let A be a 3 $$\times$$ 3 real matrix such that

$$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right);A\left( {\matrix{ 1 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right)$$ and $$A\left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 2 \cr } } \right)$$.

If $$X = {({x_1},{x_2},{x_3})^T}$$ and I is an identity matrix of order 3, then the system $$(A - 2I)X = \left( {\matrix{ 4 \cr 1 \cr 1 \cr } } \right)$$ has :