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

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Class 11th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com
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Chapter 7: Matrices and Determinants

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5
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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
Class 11th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com

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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