Using properties of determinants prove the following:

$\left| \begin{array}{ccc}1 & 1& 1\\ \mathrm{a} & \mathrm{b}& \mathrm{c}\\ {\mathrm{a}}^3 & {\mathrm{b}}^3& {\mathrm{c}}^3\end{array} \right| = \left( \mathrm{a} - \mathrm{b} \right) \left( \mathrm{b} - \mathrm{c} \right) \left( \mathrm{c} - \mathrm{a} \right) \left( \mathrm{a} + \mathrm{b} +\mathrm{c} \right)$

Using matrices solve the following system of linear equations:

x - y + 2 z = 7

3 x + 4 y - 5 z = - 5

2 x - y + 3 z = 12

If A =  and A adj A = AA^T, then 5a +b is equal to

• -1

• 5

• 4

• 4

The system of linear equations x+λy−z=0; λx−y−z=0; x+y−λz=0 has a non-trivial solution for

• infinitely many values of λ.

• exactly one value of λ.

• exactly two values of λ.

• exactly two values of λ.

A =  is a matrix satisfying the equation AA^T = 9I, Where I is 3 x 3 identity matrix, then the ordered pair (a,b) is equal to

• (2,-1)

• (-2,1)

• (2,1)

• (2,1)

The set of all values of λ for which the system of linear equations 

2x₁-2x₂+x₃ = λx₁
2x₁- 3x₂ + 2x₃ = λx₂
-x₁ + 2x₂ = λx₃
a non- trivial solution.

• is an empty set

• is a singleton set

• contains two elements

• contains two elements

If α, β ≠ 0 and f(n) = αⁿ+ βⁿ and 


= K(1-α)²(1-β)²(α- β)², then K is equal to 

• αβ 

• 1/αβ 

• 1

• 1

If A is a 3x3 non- singular matrix such that AA^T = A^TA, then BB^T is equal to

• l +B
• l

• B^-1

• B^-1

339.

Let P and Q be 3 × 3 matrices with P ≠ Q. If P³= Q³ and P²Q = Q²P, then determinant of(P²+ Q²) is equal to

• -2

• 1

• 0

• 0

The number of values of k for which the linear equations
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
posses a non-zero solution is:

• 3

• 2

• 1

• 1