Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 7 Matrices and Determinants Ex 7.5 Text Book Back Questions and Answers, Notes.

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 = \(\) (3i – 2j) and A = [aij]3 × 2 is
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 1
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 2
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 3
(4) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 4
Answer:
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 2

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

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 5
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 6

Question 2.
What must be the matrix X, if
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 7
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 8
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 9
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 10
(4)Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 11
Answer:
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 8

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

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 12

Question 3.
Which one of the following is not true about the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right] \) ?.
(1) a scalar matrix
(2) a diagonal matrix
(3) an upper triangular matrix
(4) a lower triangular matrix
Answer:
(1) a scalar matrix

Explaination:
Let A = \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right] \)
(1) a scalar matrix – not true
(2) a diagonal matrix – true
(3) an upper triangular matrix – true
(4) a lower triangular matrix – true

[(1) A square matrix A = [aij]m × n is called a diagonal matrix if aij = 0 whenever i ≠ j
(2) A diagonal matrix whose entries along the principle diagonal are equal is called a scalar matrix.
(3) A square matrix is said to be an upper triangular matrix if all the elements below the main diagonal are zero.
(4) A square matrix is said to be a lower triangular matrix if all the elements above the main diagonal are zero.]

Samacheer Kalvi 11th Maths Guide 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
(1) A and B are two matrices not necessarily of same order
(2) A and B are square matrices of same order
(3) Number of columns of A is equal to the number of rows of B
(4) A = B
Answer:
(2) A and B are square matrices of same order

Explaination:
Given A and B are two matrices such that A + B and AB are both defined.
A + B defined means A and B are same order.
AB defined means , Number of columns of A equal to Number of rows of B.
A + B and AB are simultaneously defined.
Therefore A and B are of same order.

Question 5.
If A = \(\left[ \begin{matrix} λ & 1 \\ -1 & -λ \end{matrix} \right] \), then for what value of λ, A2 = 0 ?
(1) 0
(2) ± 1
(3) – 1
(4) 1
Answer:
(2) ± 1

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 13

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

Question 6.
If A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right]\), B = \(\left[ \begin{matrix} a & 1 \\ b & -1 \end{matrix} \right]\) and (A + B)2 = A2 + B2, then the values of a and b are
(1) a = 4, b = 1
(2) a = 1, b = 4
(3) a = 0, b = 4
(4) a = 2, b = 4
Answer:
(2) a = 1, b = 4

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 14
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 15
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 16
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 17
L .2
Equating the like entrices
(a – 1)2 = a2 + b – 1 ………. (3)
o = a – 1 ………. (4)
2a+ 2 + ab + b = ab – b ………. (5)
4 = b ………. (6)
a = 1, b = 4

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

Question 7.
If A = \(\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{matrix} \right] \)
the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b)
is equal to
(1) (2, -1)
(2) (- 2, 1)
(3) (2, 1)
(4) (- 2, – 1)
Answer:
(4) (- 2, – 1)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 18
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 19
Equating the like entries
a + 2b + 4 = 0 …………. (1)
2a – 2b + 2 = 0
a – b + 1 = 0 …………. (2)
a2 + b2 + 4 = 9 …………. (3)
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 20
3b = – 3 ⇒ b = -1
Substituting in equation (1) we get
a + 2 × – 1 + 4 = 0
a – 2 + 4 = 0
a = – 2
Substituting the values a = – 2 , b = – 1 in equation (3)
we have
(- 2)2 + (-1)2 + 4 = 9
4 + 1 + 4 = 9
9 = 9
∴ The required value of the ordered pair (a, b) is
(a, b) = (- 2, – 1)

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

Question 8.
If A is a square matrix, then which of the following is not symmetric?
(1) A + AT
(2) AAT
(3) ATA
(4) AAT
Answer:
(4) AAT

Explaination:
Given A is a square matrix.
A square matrix A is symmetric if AT = A

(1) A + AT
(A + AT)T = AT + (AT)T
= AT + A = A + AT
∴ A + AT is symmetric.

(2) AAT
(AAT)T = (AT)TAT
= AAT
∴ AAT is symmetric.

(3) ATA
(ATA)T = AT(AT)T
= ATA
∴ ATA is symmetric.

(4) A – AT
(A – AT)T = AT – (AT)T
= ATA
∴ A – AT is not symmetric.

Question 9.
If A and B are symmetric matrices of order n , where (A ≠ B) , then
(1) A + B is skew – symmetric
(2) A + B is symmetric
(3) A + B is a diagonal matrix
(4) A + B is a zero matrix
Answer:
(2) A + B is symmetric

Explaination:
Given A and B are symmetric matrices of order n.
∴ AT = A and BT = B
A matrix A is skew symmetric if AT = – A
(1)(A + B)T = AT + BT = A + B
A + B is not skew symmetric.
(2)(A + B)T = AT+BT = A + B
∴ A + B is symmetric.
(3) A + B is a diagonal matrix is incorrect.
(4) A + B is a zero matrix is incorrect.

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

Question 10.
If A = \(\left[ \begin{matrix} a & x \\ y & a \end{matrix} \right] \) and if xy = 1, then det (AAT) is equal to
(1) (a – 1)2
(2) (a2 + 1)2
(3) a2 – 1
(4) (a2 – 1)2
Answer:
(4) (a2 – 1)2

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 21
= (a2 + x2) (y2 + a2) – (ax + ay) (ax + ay)
= a2 y2 + a4 + x2 y2 + a2 x2 – ((ax)2 + (ay)2 + 2 (ax)(ay))
= a2 x2 + a2 y2 + a4 + (xy)2 – a2 x2 – a2 y2 – 2a2 xy
= a4 + (1)2 – 2a2(1)
= a4 – 2a2 + 1
|AAT| = (a2 – 1)2

Question 11.
The value of x, for which the matrix A = \(\left[ \begin{matrix} { e }^{ x-2 } & { e }^{ 7+x } \\ { e }^{ 2+x } & { e }^{ 2x+3 } \end{matrix} \right] \) is singular
(1) 9
(2) 8
(3) 7
(4) 6
Answer:
(2) 8

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 22

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

Question 12.
If the points (x, – 2), (5, 2), (8, 8) are collinear , then x is equal to
(1) – 3
(2) \(\frac{1}{3}\)
(3) 1
(4) 3
Answer:
(4) 3

Explaination:
Let the given points be (x1, y1) = (x, – 2) ,
(x2, y2) = (5, 2) and (x3, y3) = (8, 8)
The condition for the three points (x1, y1), (x2, y2) and (x3, y3) to be collinear is
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 23
\(\frac{1}{2}\) [x(2 – 8) + 2(5 – 8) + 1(40 – 16)] = 0
x × – 6 + 2 × – 3 + 1 × 24 = 0
– 6x – 6 + 24 = 0
– 6x + 18 = 0
6x = 18 ⇒ x = \(\frac{18}{6}\) = 3
x = 3

Question 13.
If \(\left| \begin{matrix} 2a & { x }_{ 1 } & { y }_{ 1 } \\ 2b & { x }_{ 2 } & { y }_{ 2 } \\ 2c & { x }_{ 3 } & { y }_{ 3 } \end{matrix} \right| \) = \(\frac{\text { abc }}{2}\) ≠ 0, then the area of the triangle whose vertices are
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 24
(1) \(\frac{1}{4}\)
(2) \(\frac{1}{4}\)abc
(3) \(\frac{1}{8}\)
(4) \(\frac{1}{8}\)abc
Answer:
(3) \(\frac{1}{8}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 25
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 26
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 27

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

Question 14.
If the square of the matrix \(\left[ \begin{matrix} α & β \\ γ & -α \end{matrix} \right] \) is the unit matrix of order 2, then α, β, and γ should
(1) 1 + α2 + βγ = 0
(2) 1 – α2 – βγ = 0
(3) 1 – α2 + βγ = 0
(4) 1 + α2 – βγ = 0
Answer:
(1) 1 + α2 + βγ = 0

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 28
– α2 – βγ = 1
α2 + βγ + 1 = 0

Question 15.
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 29
(1) Δ
(2) kΔ
(3) 3 kΔ
(4) k3 Δ
Answer:
(4) k3 Δ

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 30
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 31

Question 16.
A root of the equation \(\left| \begin{matrix} 3-x & -6 & 3 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{matrix} \right| \) = 0 is
(1) 6
(2) 3
(3) 0
(4) -6
Answer:
(3) 0

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 32
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 33
0 = 0
x = 0 satisfies the given equation.
Hence the root of the given equation is x = 0.

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

Question 17.
The value of the determinant of A = \(\left[ \begin{matrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \end{matrix} \right] \) is
(1) -2 abc
(2) abc
(3) 0
(4) a2 + b2 + c2
Answer:
(3) 0

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 34
= 0 – a(0 – bc) – b (ac – 0)
= abc – abc = 0

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
(1) vertices of an equilateral triangle
(2) vertices of a right angled triangle
(3) vertices of a right angled isosceles triangle
(4) collinear
Answer:
(4) collinear

Explaination:
Given x1, x2 , x3, as well as y1, y2, y3, are in geometric progression with the same common ratio.
∴ x1 = a, x2 = ar, x3 = ar2 ,
y1 = b, y2 = br, y3 = br2
(x1, y1) = (a, b),
(x2, y2) = (ar ,br)
and (x3, y3) = (ar2, br2)
Area of the triangle whose vertices are
(x1, y1),(x2, y2) and (x3 y3) is
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 35
= \(\frac{1}{2}\)ab[1(r – r2) – 1 (r – r2) + 1 (r3 – r3)]
= \(\frac{1}{2}\)ab[r – r2 – r + r2 + 0]
= \(\frac{1}{2}\)ab × 0 = 0
∴ The points (x1, y1),(x2, y2) and (x3 y3) are collinear.

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

Question 19.
If Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 36 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 Guide Chapter 7 Matrices and Determinants Ex 7.5 37
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 38
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 39
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 40
(4) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 41
Answer:
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 38

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 42

Question 20.
If a ≠ b, b, c satisfy \(\left| \begin{matrix} a & 2b & 2c \\ 3 & b & c \\ 4 & a & b \end{matrix} \right| \) = 0 then abc =
(1) a + b + c
(2) 0
(3) b3
(4) ab + bc
Answer:
(3) b3

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 43
(a – 6) (b2 – ac) = 0
Since a ≠ 6 , we have a – 6 ≠ 0
∴ b2 – ac = 0
b2 = ac
b3 = abc

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

Question 21.
If A = \(\left| \begin{matrix} -1 & 2 & 4 \\ 3 & 1 & 0 \\ -2 & 4 & 2 \end{matrix} \right| \) and B = \(\left| \begin{matrix} -2 & 4 & 2 \\ 6 & 2 & 0 \\ -2 & 4 & 8 \end{matrix} \right| \), then B is given by
(1) B = 4A
(2) B = – 4A
(3) B = – A
(4) B = 6A
Answer:
(2) B = – 4A

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 44
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 45

Question 22.
IfA skew-symmetric of order n and C is a column matrix of order n × 1, then CT AC is
(1) an identity matrix of order n
(2) an identity matrix of order I
(3) a zero matrix of order 1
(4) an identity matrix of order 2
Answer:
(3) a zero matrix of order 1

Explaination:
Given A is a skew symmetric matrix of order n.
∴ AT = – A
C is a column matrix of order n × 1
CTis of order 1 × n
∴ CT A is of order (1 × n) × (n × n) = (1 × n)
CTAC is of order (1 × n) × (n × 1) = (1 × 1)
Since A is a skew – symmetric matrix, we have
AT = -A
(CT AC)T= CT AT (CT)T = CT (-A) C
= -CT AC
∴ CTAC is a skew – symmetric matrix.
A matrix of order 1 is skew – symmetric if A is zero matrix.
Since CTAC is a skew – symmetric and its order is 1.
∴ CTAC is a zero matrix.

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

Question 23.
The matrix A satisfying the equation
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 46
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 47
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 48
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 49
(4) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 50
Answer:
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 49

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 51
x + 3z = 1 ——– (1)
y + 3t = 1 ——- (2)
z = 0
t = – 1
(1) ⇒ x + 3 × 0 = 1 ⇒ x = 1
(2) ⇒ y + 3 × – 1 = 1 ⇒ y = 4
∴ A = \(\left[ \begin{matrix} 1 & 4 \\ 0 & -1 \end{matrix} \right] \)

Question 24.
If A + 1 = \(\left[ \begin{matrix} 3 & -2 \\ 4 & 1 \end{matrix} \right]\), then (A + I) (A – I) is equal to
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 52
(2) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 53
(3) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 54
(4) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 55
Answer:
(1) Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 52

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 56
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.5 57

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

Question 25.
Let A and B be two symmetrh matrices of same order. Then which one of the following statement is not true?
(1) A + B is a symmetric matrix
(2) AB is a symmetric matrix
(3) AB = (BA)T
(4) ATB = MIT
Answer:
(2) AB is a symmetric matrix

Explaination:
Given A and B are two symmetric matrices of the same order.
A = AT , B = BT
(1)(A+B)T = AT + BT = A + B
A + B is symmetric.

(2) (AB)T = BT AT BA
Thus (AB)T ≠ AB
Hence , AB is not symmetric.

(3) AB = (BA)T
= AT BT = AB
Statement is true.

(4) AT B = ABT Since AT = A
B = BT
Statement is true.