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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Choose the correct or the most suitable answer from the given four alternatives.

Question 1.
If aij = \(\frac{1}{2}\) (3i – 2j) and A = [aij]2×2 is
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 2

Question 2.
What must be the matrix X, if 2X + \(\left[\begin{array}{ll}{1} & {2} \\ {3} & {4}\end{array}\right]=\left[\begin{array}{ll}{3} & {8} \\ {7} & {2}\end{array}\right]\) ?
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 3
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 4
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 5

Question 3.
Which one of the following is not true about the matrix \(\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {0} \\ {0} & {0} & {5}\end{array}\right]\)?
(a) a scalar matrix
(b) a diagonal matrix
(c) an upper triangular matrix
(d) A lower triangular matrix
Solution:
(b) a diagonal matrix

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Question 4.
If A and B are two matrices such that A + B and AB are both defined, then …………
(a) A and B are two matrices not necessarily of same order.
(b) A and B are square matrices of same order.
(c) Number of columns of a is equal to the number of rows of B.
(d) A = B.
Solution:
(b) A and B are square matrices of same order.

Question 5.
If A = \(\left[\begin{array}{rr}{\lambda} & {1} \\ {-1} & {-\lambda}\end{array}\right]\), then for what value of λ, A2 = 0?
(a) 0
(b) ±1
(c) -1
(d) 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 6

Question 6.
If Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 7 and (A + B)2 = A2 + B2, then the values of a and b are ……………….
(a) a = 4, b = 1
(b) a = 1, b = 4
(c) a = 0, b = 4
(d) a = 2, b = 4
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 8
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 9

Question 7.
If Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 10 is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to ………….
(a) (2, -1)
(b) (-2, 1)
(c) (2, 1)
(d) (-2, -1)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 11
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 12

Question 8.
If A is a square matrix, then which of the following is not symmetric?
(a) A + AT
(b) AAT
(c) ATA
(d)A – AT
Solution:
(b)

Question 9.
If A and B are symmetric matrices of order n, where (A ≠ B), then …………….
(a) A + B is skew-symmetric
(b) A + B is symmetric
(c) A + B is a diagonal matrix
(d) A + B is a zero matrix
Solution:
(b)

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Question 10.
If Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 13 and if xy = 1, then det (AAT) is equal to …………..
(a) (a – 1)2
(b) (a2 + 1)2
(c) a2 – 1
(d) (a2 – 1)2
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 14

Question 11.
The value of x, for which the matrix Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 15is singular is ………….
(a) 9
(b) 8
(c) 7
(d) 6
Solution:
(b) Hint: Given A is a singular matrix ⇒ |A| = 0
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 16
⇒ ex-2.e2x+3 – e2+x.e7+x = 0
⇒ e3x+1 – e9+2x = 0 ⇒ e3x+1 = e9+2x
⇒ 3x + 1 = 9 + 2x
3x – 2x = 9 – 1 ⇒ x = 8

Question 12.
If the points (x, -2), (5, 2), (8, 8) are collinear, then x is equal to …………
(a) -3
(b) \(\frac{1}{3}\)
(c) 1
(d) 3
Solution:
(d) Hint: Given that the points are collinear
So, area of the triangle formed by the points = 0
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 17

Question 13.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 18
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 19
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 20

Question 14.
If the square of the matrix Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 21 is the unit matrix of order 2, then α, β and γ should satisfy the relation.
(a) 1 + α2 + βγ = 0
(b) 1 – α2 – βγ = 0
(c) 1 – α2 + βγ = 0
(d) 1 + α2 – βγ = 0
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 22

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Question 15.
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 23
(a) Δ
(b) kΔ
(c) 3kΔ
(d) k3Δ
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 24

Question 16.
A root of the equation Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 25 is …………….
(a) 6
(b) 3
(c) 0
(d) -6
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 26

Question 17.
The value of the determinant of Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 27is ……………
(a) -2abc
(b) abc
(c) 0
(d) a2 + b2 + c2
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 28

Question 18.
If x1, x2, x3 as well as y1, y2, y3 are in geometric progression with the same common ratio, then the points (x1, y1), (x2, y2), (x3, y3) are
(a) vertices of an equilateral triangle
(b) vertices of a right angled triangle
(c) vertices of a right angled isosceles triangle
(d) collinear
Solution:
(d)

Question 19.
If \(\lfloor.\rfloor\) denotes the greatest integer less than or equal to the real number under consideration and -1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 29 is …………..
(a) \(\lfloor z\rfloor\)
(b) \(\lfloor y\rfloor\)
(c) \(\lfloor x\rfloor\)
(d) \(\lfloor x\rfloor+ 1\)
Solution:
(a) Hint: From the given values
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 30>

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Question 20.
If a ≠ b, b, c satisfy Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 31 then abc = ……………..
(a) a + b + c
(b) 0
(c) b3
(d) ab + bc
Solution:
(c) Hint: Expanding along R1,
a(b2 – ac) – 2b (3b – 4c) + 2c (3a – 4b) = 0
(b2 – ac) (a – b) = 0
b2 = ac (or) a = b
⇒ abc = b(b2) = b3

Question 21.
If Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 32 then B is given by ………………..
(a) B = 4A
(b) B = -4A
(c) B = -A
(d) B = 6A
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 33
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 34

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Question 22.
IfA is skew-symmetric of order n and C ¡s a column matrix of order n × 1, then CT AC is ……………..
(a) an identity matrix of order n
(b) an identity matrix of order 1
(e) a zero matrix of order I
(d) an Identity matrix of order 2
Solution:
(c) Hint : Given A is of order n × n
C is of order n × 1
so, CT is of order 1 × n
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 35
Let it be equal to (x) say
Taking transpose on either sides
(CT, AC)T (x)T .
(i.e.) CT(AT)(C) = x
CT(-A)(C) = x
⇒ CTAC = -x
⇒ x = -x ⇒ 2x = 0 ⇒ x = 0

Question 23.
The matrix A satisfying the equation Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 36 is ……………
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 37
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 38

Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 39

Question 24.
If A + I = Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 40, then (A + I) (A – I) is equal to …………….
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 41
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5 42

Question 25.
Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?
(a) A + B ¡s a symmetric matrix
(b) AB ¡s a symmetric matrix
(c) AB = (BA)T
(d) ATB = ABT
Solution:
(b)

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