Matrices and Determinants

1

If  then adj (3A2 + 12A) is equal to

2

Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

  1. 2, -1/2
  2. 1, 3/4
  3. 1, -3/4
  4. 2, 1/2
3

If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is:

  1. an empty set
  2. an infinite set
  3. a finite set containing two or more elements
  4. a singleton
4

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a non-­zero solution, then a, b, c

  1. satisfy a + 2b + 3c
  2. are in G.P
  3. are in A.P
  4. are in H.P
5

If A and B are square matrices of size n × n such that A2 – B2 = (A – B)(A + B), then which of the following will be always true?

  1. either A or B is an identity matrix
  2. either A or B is a zero matrix
  3. AB = BA
  4. A = B
6

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

  1. If det A = ± 1, then A-1 exists and all its entries are integers
  2. If det A = ± 1, then A-1 exists and all its entries are non-integers
  3. If det A = ± 1, then A-1 exists and all its entries are not necessarily integers
  4. If det A = ± 1, then A-1 need not exist
7

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

  1. 0
  2. 1
  3. -1
  4. -2
8

If  A2 – A + I = 0, then the inverse of A is

  1. A - I
  2. A
  3. I + A
  4. I - A
9

If \( A = \begin{bmatrix}1 & 2 \\2 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix}1 & 0 \\1 & 0 \end{bmatrix} \) then what is determinant of AB?

  1. 0
  2. 1
  3. 10
  4. 20
10

A and B are two matrices such that AB = A and BA = B then what is B2 equal to?

  1. B
  2. A
  3. I
  4. -I
11

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

  1. exactly one value of λ.
  2. exactly two values of λ.
  3. exactly three values of λ.
  4. infinitely many values of λ.