#### Online Mock Tests

#### Chapters

Chapter 2: Basic Algebra

Chapter 3: Trigonometry

Chapter 4: Combinatorics and Mathematical Induction

Chapter 5: Binomial Theorem, Sequences and Series

Chapter 6: Two Dimensional Analytical Geometry

Chapter 7: Matrices and Determinants

Chapter 8: Vector Algebra

Chapter 9: Differential Calculus - Limits and Continuity

Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation

Chapter 11: Integral Calculus

Chapter 12: Introduction to probability theory

## Chapter 7: Matrices and Determinants

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

Construct an m × n matrix A = [a_{ij}], where a_{ij} is given by

a_{ij} = `("i" - 2"j")^2/2` with m = 2, n = 3

Construct an m × n matrix A = [a_{ij}], where a_{ij} is given by

a_{ij} = `|3"i" - 4"j"|/4` with m = 3, n = 4

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)]`

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

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)]`

If A = `[(1, "a"),(0, 1)]`, then compute A^{4}

Consider the matrix A_{α} = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Show that `"A"_alpha "A"_beta = "A"_((alpha + beta))`

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

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

If A = `[(1, 0, 0),(0, 1, 0),("a", "b", - 1)]`, show that A^{2} is a unit matrix

If A = `[(1, 0, 2), (0, 2, 1), (2, 0, 3)]` and A^{3} – 6A^{2} + 7A + kI = 0, find the value of k

Give your own examples of matrices satisfying the following conditions:

A and B such that AB ≠ BA

Give your own examples of matrices satisfying the following conditions:

A and B such that AB = 0 = BA, A ≠ 0 and B ≠ 0

Give your own examples of matrices satisfying the following conditions:

A and B such that AB = 0 and BA ≠ 0

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

If A is a square matrix such that A^{2} = A, find the value of 7A – (I + A)^{3}

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)]`

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

If A^{T} = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following

(A + B)^{T} = A^{T} + B^{T} = B^{T} + A^{T}

If A^{T} = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following

(A – B)^{T} = A^{T} – B^{T}

If A^{T} = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following

(B^{T})^{T} = B

If A is a 3 × 4 matrix and B is a matrix such that both A^{T}B and BA^{T} are defined, what is the order of the matrix B?

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

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

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)]`

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

If A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]` is a matrix such that AA^{T} = 9I, find the values of x and y

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

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

Construct the matrix A = [a_{ij}]_{3×3}, where a_{ij} = 1 – j. State whether A is symmetric or skew–symmetric

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

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

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

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?

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

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

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

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`

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

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

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

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

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 ax^{2} + 2bx + c = 0

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

If a, b, c are p^{th}, q^{th} and r^{th} terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`

Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x^{4 }

If a, b, c, are all positive, and are p^{th}, q^{th} and r^{th} terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0

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

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

Without expanding, evaluate the following determinants:

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

Without expanding, evaluate the following determinants:

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

If A is a Square, matrix, and |A| = 2, find the value of |A A^{T}|

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

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

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

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)]`

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Matrices and Determinants Exercise 7.3 [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)

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

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

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)

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

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

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

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

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

Identify the singular and non-singular matrices:

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

Identify the singular and non-singular matrices:

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

Identify the singular and non-singular matrices:

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

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

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

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)]`

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

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)|`

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

Choose the correct alternative:

If a_{ij} = (3i – 2j) and A = [a_{ij}]_{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)]`

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)]`

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

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

Choose the correct alternative:

if A = `[(lambda, 1),(-1, -lambda)]`, then for what value of λ, A^{2} = 0 ?

0

± 1

– 1

1

Choose the correct alternative:

If A = `[(1, -1),(2, -1)]`, B = `[("a", 1),("b", -1)]` and (A + B)^{2} = A^{2} + B^{2}, then the values of a and b are

a = 4, b = 1

a = 1, b = 4

a = 0, b = 4

a = 2, b = 4

Choose the correct alternative:

If A = `[(1, 2, 2),(2, 1, -2),("a", 2, "b")]` is a matrix satisfying the equation AA^{T} = 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)

Choose the correct alternative:

If A is a square matrix, then which of the following is not symmetric?

A + A

^{T}AA

^{T}A

^{T}AA – A

^{T}

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

Choose the correct alternative:

If A = `[("a", x),(y, "a")]` and if xy = 1, then det(AA^{T}) is equal to

(a – 1)

^{2}(a

^{2}+ 1)^{2}a

^{2}– 1(a

^{2}– 1)^{2}

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

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

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"`

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 + α

^{2}+ βγ = 01 – α

^{2}– βγ = 01 – α

^{2}+ βγ = 01 + α

^{2}– βγ = 0

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

kΔ

3kΔ

k

^{3}Δ

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

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

a

^{2}+ b^{2}+ c^{2}

Choose the correct alternative:

If x_{1}, x_{2}, x_{3} as well as y_{1}, y_{2}, y_{3} are in geometric progression with the same common ratio, then the points (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) are

vertices of an equilateral triangle

vertices of a right angled triangle

vertices of a right angled isosceles triangle

collinear

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

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

b

^{3}ab + bc

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

Choose the correct alternative:

If A is skew-symmetric of order n and C is a column matrix of order n × 1, then C^{T} 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

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)]`

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)]`

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}A

^{T}B = MI^{T}

## Chapter 7: Matrices and Determinants

## 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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Concepts covered in Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 7 Matrices and Determinants are Introduction to Matrices and Determinants, Matrices, Determinants.

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