**Question 16: For what value of p the following set of equations will have no solution? [2015 : 1 Mark, Set-I]**

2x + 3y = 5

3x + py = 10

Solution: Given system of equations has no solution if the lines are parallel i.e., their slopes are equal

2/3 = 3/p

⇒ p = 4.5

**Question 17: The rank of the matrix is _____. [2014 : 2 Marks, Set-II]**

**Solution:**

Determinant of matrix is not zero.

∴ Rank is 2

**Question 18: The determinant of matrix [2014 : 1 Mark, Set-II]**

**Solution: **

Interchanging column 1 and column 2 and taking transpose,

**Question 19: With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x**_{1}, y_{1}) = (1, 0); (x_{2}, y_{2}) = (2, 2); (x_{3}, y_{3}) = (4, 3). The area of the triangle is equal to [2014 : 1 Mark, Set-I]

**(a) 3/2**

**(b) 3/4**

**(c) 4/5**

**(d) 5/2**

**Answer: (a)**

**Solution: **

Area of triangle is

**Question 20: The sum of Eigen values of matrix, [M] is where [2014 : 1 Mark, Set-I]**

**(a) 915 **

**(b) 1355 **

**(c) 1640 **

**(d) 2180**

**Answer:** (a)

**Solution:** Sum of eigen values = trace of matrix

= 215 + 150 + 550 = 915

**Question 21: Given the matrices the product K**^{T} JK is ____. [2014 : 1 Mark, Set-I]

**Solution:**

**Question 22: There are three matrixes P(4 x 2), Q(2 x 4) and R(4 x 1). The minimum of multiplication required to compute the matrix PQR is [2013 : 1 Mark]**

**Solution: **If we multiply QR first then,

Q_{2x4} x R_{(4x1)} having multiplication number 8.

There fore P_{(4 x 2)} QR_{(2 x 1)} will have minimum number of multiplication = (8 + 8) = 16.

**Question 23: The eigen values of matrix [2011 : 2 Marks] **

**(a) -2.42 and 6.86 **

**(b) 3.48 and 13.53 **

**(c) 4.70 and 6.86 **

**(d) 6.86 and 9.50**

**Answer: **(b)

**Solution: **We need eigen values of

The characteristic equation is,

So eigen values are,

λ = 3.48, 13.53

**Question 24: [A] is square matix which is neither symmetric nor skew-symmetric and [A]**^{T} is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]^{T} and [D] = [A] - [A]^{T}, respectively. Which of the following statements is TRUE? [2011 : 1 Mark]

**(a) Both [S] and [D] are symmetric**

**(b) Both [S] and [D] are skew-symmetric **

**(c) [S] is skew-symmetric and [D] is symmetric **

**(d) [S] is symmetric and [D] is skew-symmetric**

**Answer:** (d)

**Solution: **Since (A + A^{t}) = A^{t} + (A^{t})^{t}

= A^{t} + A

i.e. S^{t} = S

∴ S is symmetric

Since (A - A^{t})^{t} = A^{t} - (A^{t})^{t}

= A^{t} - A = -(A - A^{t})

i.e. D^{t} = - D

So D is Skew-Symmetric.

**Question 25: The inverse of the matrix [2010 : 2 Marks]**

**Answer: (b)**

**Solution:**