Tamil Nadu Board of Secondary EducationHSC Science Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Matrices and Determinants [Latest edition]

#### Chapters ## Chapter 7: Matrices and Determinants

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5
Exercise 7.1 [Pages 17 - 19]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and DeterminantsExercise 7.1 [Pages 17 - 19]

Exercise 7.1 | Q 1. (i) | Page 17

Construct an m × n matrix A = [aij], where aij is given by

aij = ("i" - 2"j")^2/2 with m = 2, n = 3

Exercise 7.1 | Q 1. (ii) | Page 17

Construct an m × n matrix A = [aij], where aij is given by

aij = |3"i" - 4"j"|/4 with m = 3, n = 4

Exercise 7.1 | Q 2 | Page 18

Find the values of p, q, r, and s if

[("p"^2 - 1, 0, - 31 - "q"^3),(7, "r" + 1, 9),(- 2, 8, "s" - 1)] = [(1, 0, -4),(7, 3/2, 9),(-2, 8, -pi)]

Exercise 7.1 | Q 3 | Page 18

Determine the value of x + y if [(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)]

Exercise 7.1 | Q 4 | Page 18

Determine the matrices A and B if they satisfy 2A – B + [(6, - 6, 0),(- 4, 2, 1)] = 0 and A – 2B = [(3, 2, 8),(-2, 1, -7)]

Exercise 7.1 | Q 5 | Page 18

If A = [(1, "a"),(0, 1)], then compute A4

Exercise 7.1 | Q 6. (i) | Page 18

Consider the matrix Aα = [(cos alpha, - sin alpha),(sin alpha, cos alpha)] Show that "A"_alpha "A"_beta = "A"_((alpha + beta))

Exercise 7.1 | Q 6. (ii) | Page 18

Consider the matrix Aα = [(cos alpha, - sin alpha),(sin alpha, cos alpha)] Find all possible real values of α satisfying the condition "A"_alpha + "A"_alpha^"T" = I

Exercise 7.1 | Q 7 | Page 18

If A = [(4, 2),(-1, x)] and such that (A – 2I)(A – 3I) = 0, find the value of x

Exercise 7.1 | Q 8 | Page 18

If A = [(1, 0, 0),(0, 1, 0),("a", "b", - 1)], show that A2 is a unit matrix

Exercise 7.1 | Q 9 | Page 18

If A = [(1, 0, 2), (0, 2, 1), (2, 0, 3)] and A3 – 6A2 + 7A + kI = 0, find the value of k

Exercise 7.1 | Q 10. (i) | Page 18

Give your own examples of matrices satisfying the following conditions:
A and B such that AB ≠ BA

Exercise 7.1 | Q 10. (ii) | Page 18

Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 = BA, A ≠ 0 and B ≠ 0

Exercise 7.1 | Q 10. (iii) | Page 18

Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 and BA ≠ 0

Exercise 7.1 | Q 11 | Page 18

Show that f(x) f(y) = f(x + y), where f(x) = [(cosx, -sinx, 0),(sinx, cosx, 0),(0, 0, 1)]

Exercise 7.1 | Q 12 | Page 18

If A is a square matrix such that A2 = A, find the value of 7A – (I + A)3

Exercise 7.1 | Q 13 | Page 18

Verify the property A(B + C) = AB + AC, when the matrices A, B, and C are given by A = [(2, 0, -3),(1, 4, 5)], B = [(3, 1),(-1, 0),(4, 2)] and C = [(4, 7),(2, 1),(1,-1)]

Exercise 7.1 | Q 14 | Page 19

Find the matrix A which satisfies the matrix relation "A"= [(1, 2, 3),(4, 5, 6)] = [(-7, -8, -9),(2, 4, 6)]

Exercise 7.1 | Q 15. (i) | Page 19

If AT = [(4, 5),(-1, 0),(2, 3)] and B = [(2, -1, 1),(7, 5, -2)], veriy the following

(A + B)T = AT + BT = BT + AT

Exercise 7.1 | Q 15. (ii) | Page 19

If AT = [(4, 5),(-1, 0),(2, 3)] and B = [(2, -1, 1),(7, 5, -2)], veriy the following

(A – B)T = AT – BT

Exercise 7.1 | Q 15. (iii) | Page 19

If AT = [(4, 5),(-1, 0),(2, 3)] and B = [(2, -1, 1),(7, 5, -2)], veriy the following

(BT)T = B

Exercise 7.1 | Q 16 | Page 19

If A is a 3 × 4 matrix and B is a matrix such that both ATB and BAT are defined, what is the order of the matrix B?

Exercise 7.1 | Q 17. (i) | Page 19

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

[(4, -2),(3, -5)]

Exercise 7.1 | Q 17. (ii) | Page 19

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]

Exercise 7.1 | Q 18 | Page 19

Find the matrix A such that [(2, -1),(1, 0),(-3, 4)]"A"^"T" = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]

Exercise 7.1 | Q 19 | Page 19

If A = [(1, 2, 2),(2, 1, -2),(x, 2, y)] is a matrix such that AAT = 9I, find the values of x and y

Exercise 7.1 | Q 20. (i) | Page 19

For what value of x, the matrix A = [(0, 1, -2),(-1, 0, x^3),(2, -3, 0)] is skew – symmetric

Exercise 7.1 | Q 20. (ii) | Page 19

If [(0, "p", 3),(2, "q"^2, -1),("r", 1, 0)] is skew – symmetric find the values of p, q and r

Exercise 7.1 | Q 21 | Page 19

Construct the matrix A = [aij]3×3, where aij = 1 – j. State whether A is symmetric or skew–symmetric

Exercise 7.1 | Q 22 | Page 19

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix

Exercise 7.1 | Q 23. (i) | Page 19

If A and B are symmetric matrices of same order, prove that AB + BA is a symmetric matrix

Exercise 7.1 | Q 23. (ii) | Page 19

If A and B are symmetric matrices of same order, prove that AB – BA is a skew-symmetric matrix

Exercise 7.1 | Q 24 | Page 19

A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds. Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds. Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds. Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds. The cost of 50 gm of cashew nuts is ₹ 50, 50 gm of raisins is ₹ 10, and 50 gm of almonds is ₹ 60. What is the cost of each gift pack?

Exercise 7.2 [Pages 28 - 30]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and DeterminantsExercise 7.2 [Pages 28 - 30]

Exercise 7.2 | Q 1 | Page 28

Without expanding the determinant, prove that |("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)| = 0

Exercise 7.2 | Q 2 | Page 28

Show that |("b" + "c", "bc", "b"^2"C"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)| = 0

Exercise 7.2 | Q 3 | Page 29

Prove that |("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| = 4"a"^2"b"^2"c"^2

Exercise 7.2 | Q 4 | Page 29

Prove that |(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")

Exercise 7.2 | Q 5 | Page 29

Prove that |(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)| = 0

Exercise 7.2 | Q 6 | Page 29

Show that |(x + 2"a", y + 2"b", z + 2"c"),(x, y, z),("a", "b", "c")| = 0

Exercise 7.2 | Q 7 | Page 29

Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0

Exercise 7.2 | Q 8 | Page 29

If |("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)| = 0, prove that a, b, c are in G. P or α is a root of ax2 + 2bx + c = 0

Exercise 7.2 | Q 9 | Page 29

Prove that |(1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")| = 0

Exercise 7.2 | Q 10 | Page 29

If a, b, c are pth, qth and rth terms of an A.P, find the value of |("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|

Exercise 7.2 | Q 11 | Page 29

Show that |("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)| is divisiible by x

Exercise 7.2 | Q 12 | Page 29

If a, b, c, are all positive, and are pth, qth and rth terms of a G.P., show that |(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)| = 0

Exercise 7.2 | Q 13 | Page 29

Find the value of |(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)| if x, y, z ≠ 1

Exercise 7.2 | Q 14 | Page 30

If A = [(1/2, alpha),(0, 1/2)], prove that sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)

Exercise 7.2 | Q 15. (i) | Page 30

Without expanding, evaluate the following determinants:

|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|

Exercise 7.2 | Q 15. (ii) | Page 30

Without expanding, evaluate the following determinants:

|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|

Exercise 7.2 | Q 16 | Page 30

If A is a Square, matrix, and |A| = 2, find the value of |A AT|

Exercise 7.2 | Q 17 | Page 30

If A and B are square matrices of order 3 such that |A| = –1 and |B| = 3, find the value of |3AB|

Exercise 7.2 | Q 18 | Page 30

If λ = – 2, determine the value of |(0, lambda, 1),(lambda^2, 0, 3lambda^2 + 1),(-1, 6lambda - 1, 0)|

Exercise 7.2 | Q 19 | Page 30

Determine the roots of the equation |(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)| = 0

Exercise 7.2 | Q 20 | Page 30

Verify that det(AB) = (det A)(det B) for A = [(4, 3, -2),(1, 0, 7),(2, 3, -5)] and B = [(1, 3, 3),(-2, 4, 0),(9, 7, 5)]

Exercise 7.2 | Q 21 | Page 30

Using cofactors of elements of second row, evaluate |A|, where A = [(5, 3, 8),(2, 0, 1),(1, 2, 3)]

Exercise 7.3 [Page 34]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and DeterminantsExercise 7.3 [Page 34]

Exercise 7.3 | Q 1 | Page 34

Solve the following problems by using Factor Theorem:

Show that |(x, "a", "a"),("a", x, "a"),("a", "a", x)| = (x – a)2 (x + 2a)

Exercise 7.3 | Q 2 | Page 34

Show that |("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")| = 8abc

Exercise 7.3 | Q 3 | Page 34

Solve that |(x + "a", "b", "c"),("a", x + "b", "c"),("a", "b", x + "c")| = 0

Exercise 7.3 | Q 4 | Page 34

Show that |("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)| = (a + b + c)(a – b)(b – c)(c – a)

Exercise 7.3 | Q 5 | Page 34

Solve |(4 - x, 4 + x, 4 +  x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)| = 0

Exercise 7.3 | Q 6 | Page 34

Show that |(1, 1, 1),(x, y, z),(x^2, y^2, z^2)| = (x – y)(y – z)(z – x)

Exercise 7.4 [Pages 39 - 40]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and DeterminantsExercise 7.4 [Pages 39 - 40]

Exercise 7.4 | Q 1 | Page 39

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)

Exercise 7.4 | Q 2 | Page 39

If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k

Exercise 7.4 | Q 3. (i) | Page 39

Identify the singular and non-singular matrices:

[(1, 2, 3),(4, 5, 6),(7, 8, 9)]

Exercise 7.4 | Q 3. (ii) | Page 39

Identify the singular and non-singular matrices:

[(2, -3, 5),(6, 0, 4),(1, 5, -7)]

Exercise 7.4 | Q 3. (iii) | Page 39

Identify the singular and non-singular matrices:

[(0, "a" - "b", "k"),("b" - "a", 0, 5),(-"k", -5, 0)]

Exercise 7.4 | Q 4. (i) | Page 40

Determine the values of a and b so that the following matrices are singular:

A = [(7, 3),(-2, "a")]

Exercise 7.4 | Q 4. (ii) | Page 40

Determine the values of a and b so that the following matrices are singular:

B = [("b" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]

Exercise 7.4 | Q 5 | Page 40

If cos 2θ = 0, determine  [(theta, costheta, sintheta),(costheta, sintheta, 0),(sintheta, 0, costheta)]^2

Exercise 7.4 | Q 6 | Page 40

Find the value of the product: |(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)|

Exercise 7.5 [Pages 40 - 43]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and DeterminantsExercise 7.5 [Pages 40 - 43]

Exercise 7.5 | Q 1 | Page 40

Choose the correct alternative:
If aij =  (3i – 2j) and A = [aij]3 × 2 is

• [(1/2, 2),(-1/2, 1)]

• [(1/2, -1/2),(2, 1)]

• [(2, 2),(1/2, -1/2)]

• [(-1/2, 1/2),(1, 2)]

Exercise 7.5 | Q 2 | Page 40

Choose the correct alternative:
What must be the matrix X, if 2"X" + [(1, 2),(3, 4)] = [(3, 8),(7, 2)]?

• [(1, 3),(2, -1)]

• [(1, -3),(2, -1)]

• [(2, 6),(4, -2)]

• [(2, -6),(4, -2)]

Exercise 7.5 | Q 3 | Page 40

Choose the correct alternative:
Which one of the following is not true about the matrix [(1, 0, 0),(0, 0, 0),(0, 0, 5)]?

• a scalar matrix

• a diagonal matrix

• an upper triangular matrix

• a lower triangular matrix

Exercise 7.5 | Q 4 | Page 40

Choose the correct alternative:
If A and B are two matrices such that A + B and AB are both defined, then

• A and B are two matrices not necessarily of same order

• A and B are square matrices of same order

• Number of columns of A is equal to the number of rows of B

• A = B

Exercise 7.5 | Q 5 | Page 40

Choose the correct alternative:
if A = [(lambda, 1),(-1, -lambda)], then for what value of λ, A2 = 0 ?

• 0

• ± 1

• – 1

• 1

Exercise 7.5 | Q 6 | Page 41

Choose the correct alternative:
If A = [(1, -1),(2, -1)], B = [("a", 1),("b", -1)] and (A + B)2 = A2 + B2, then the values of a and b are

• a = 4, b = 1

• a = 1, b = 4

• a = 0, b = 4

• a = 2, b = 4

Exercise 7.5 | Q 7 | Page 41

Choose the correct alternative:
If A = [(1, 2, 2),(2, 1, -2),("a", 2, "b")] is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

• (2, – 1)

• (– 2, 1)

• (2, 1)

• (– 2, – 1)

Exercise 7.5 | Q 8 | Page 41

Choose the correct alternative:
If A is a square matrix, then which of the following is not symmetric?

• A + AT

• AAT

• ATA

• A – AT

Exercise 7.5 | Q 9 | Page 41

Choose the correct alternative:
If A and B are symmetric matrices of order n, where (A ≠ B), then

• A + B is skew-symmetric

• A + B is symmetric

• A + B is a diagonal matrix

• A + B is a zero matrix

Exercise 7.5 | Q 10 | Page 41

Choose the correct alternative:
If A = [("a", x),(y, "a")] and if xy = 1, then det(AAT) is equal to

• (a – 1)2

• (a2 + 1)2

• a2 – 1

• (a2 – 1)2

Exercise 7.5 | Q 11 | Page 41

Choose the correct alternative:
The value of x, for which the matrix A = [("e"^(x - 2), "e"^(7 + x)),("e"^(2 + x), "e"^(2x + 3))] is singular

• 9

• 8

• 7

• 6

Exercise 7.5 | Q 12 | Page 41

Choose the correct alternative:
If the points (x, – 2), (5, 2), (8, 8) are collinear, then x is equal to

• – 3

• 1/3

• 1

• 3

Exercise 7.5 | Q 13 | Page 41

Choose the correct alternative:
If |(2"a", x_1, y_1),(2"b", x_2, y_2),(2"c", x_3, y_3)| = "abc"/2 ≠ 0, then the area of the triangle whose vertices are (x_1/"a", y_1/"a"), (x_2/"b", y_2/"b"), (x_3/"c", y_3/"c") is

• 1/4

• 1/4 "abc"

• 1/8

• 1/8 "abc"

Exercise 7.5 | Q 14 | Page 41

Choose the correct alternative:
If the square of the matrix [(alpha, beta),(γ, - alpha)] is the unit matrix of order 2, then α, β, and γ should

• 1 + α2 + βγ = 0

• 1 – α2 – βγ = 0

• 1 – α2 + βγ = 0

• 1 + α2 – βγ = 0

Exercise 7.5 | Q 15 | Page 42

Choose the correct alternative:
if Δ = |("a", "b", "c"),(x, y, z),("p", "q", "r")| then  |("ka", "kb","kc"),("k"x, "k"y, "k"z),("kp", "kq", "kr")| is

• Δ

• 3kΔ

• k3Δ

Exercise 7.5 | Q 16 | Page 42

Choose the correct alternative:
A root of the equation |(3 - x, -6, 3),(-6, 3 - x, 3),(3, 3, -6 - x)| = 0 is

• 6

• 3

• 0

• – 6

Exercise 7.5 | Q 17 | Page 42

Choose the correct alternative:
The value of the determinant of A = [(0, "a", -"b"),(-"a", 0, "c"),("b", -"c", 0)] is

• – 2 abc

• abc

• 0

• a2 + b2 + c2

Exercise 7.5 | Q 18 | Page 42

Choose the correct alternative:
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

• vertices of an equilateral triangle

• vertices of a right angled triangle

• vertices of a right angled isosceles triangle

• collinear

Exercise 7.5 | Q 19 | Page 42

Choose the correct alternative:
If ⌊.⌋ 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 [([x] + 1, [y], [z]),([x], [y] + 1, [z]),([x], [y], [z] + 1)]

• [z]

• [y]

• [x]

• [x] + 1

Exercise 7.5 | Q 20 | Page 42

Choose the correct alternative:
If a ≠ b, b, c satisfy |("a", 2"b", 2"c"),(3, "b", "c"),(4, "a", "b")| = 0, then abc =

• a + b + c

• 0

• b3

• ab + bc

Exercise 7.5 | Q 21 | Page 42

Choose the correct alternative:
If A = |(-1, 2, 4),(3, 1, 0),(-2, 4, 2)| and B = |(-2, 4, 2),(6, 2, 0),(-2, 4, 8)|, then B is given by

• B = 4A

• B = – 4A

• B = – A

• B = 6A

Exercise 7.5 | Q 22 | Page 43

Choose the correct alternative:
If A is skew-symmetric of order n and C is a column matrix of order n × 1, then CT AC is

• an identity matrix of order n

• an identity matrix of order 1

• a zero matrix of order 1

• an identity matrix of order 2

Exercise 7.5 | Q 23 | Page 43

Choose the correct alternative:
The matrix A satisfying the equation [(1, 3),(0, 1)] "A" = [(1, 1),(0, -1)] is

• [(1, 4),(-1, 0)]

• [(1, -4),(1, 0)]

• [(1, 4),(0, -1)]

• [(1, -4),(1, 1)]

Exercise 7.5 | Q 24 | Page 43

Choose the correct alternative:
If A + I = [(3, -2),(4, 1)], then (A + I)(A – I) is equal to

• [(-5, -4),(8, -9)]

• [(-5, 4),(-8, 9)]

• [(5, 4),(8, 9)]

• [(-5, -4),(-8, -9)]

Exercise 7.5 | Q 25 | Page 43

Choose the correct alternative:
Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?

• A + B is a symmetric matrix

• AB is a symmetric matrix

• AB = (BA)T

• ATB = MIT

## Chapter 7: Matrices and Determinants

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5 ## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Matrices and Determinants

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