# NCERT solutions for Mathematics Exemplar Class 12 chapter 3 - Matrices [Latest edition]

## Chapter 3: Matrices

Solved ExamplesExercise
Solved Examples [Pages 46 - 52]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 3 MatricesSolved Examples [Pages 46 - 52]

Solved Examples | Q 1 | Page 46

Construct a matrix A = [aij]2×2 whose elements aij are given by aij = e2ix sin jx.

Solved Examples | Q 2 | Page 46

If A = [(2, 3),(1, 2)], B = [(1, 3, 2),(4, 3, 1)], C = [(1),(2)], D = [(4, 6, 8),(5, 7, 9)], then which of the sums A + B, B + C, C + D and B + D is defined?

Solved Examples | Q 3 | Page 46

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Solved Examples | Q 4 | Page 47

If [(2x, 3)] [(1, 2),(-3, 0)] [(x),(8)] = 0, find the valof x.

Solved Examples | Q 5 | Page 47

If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero), kA is invertible and ("kA")^-1 = 1/"k" "A"^-1

Solved Examples | Q 6 | Page 47

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = [(2, 4, -6),(7, 3, 5),(1, -2, 4)]

Solved Examples | Q 7 | Page 48

If A = [(1, 3, 2),(2, 0, -1),(1, 2, 3)], then show that A satisfies the equation A3 – 4A2 – 3A + 11I = O.

Solved Examples | Q 8 | Page 50

Let A = [(2, 3),(-1, 2)]. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.

#### Objective Type Questions Examples 9 to 12

Solved Examples | Q 9 | Page 51

If A and B are square matrices of the same order, then (A + B)(A – B) is equal to ______.

• A2 – B2

• A2 – BA – AB – B2

• A2 – B2 + BA – AB

• A2 – BA + B2 + AB

Solved Examples | Q 10 | Page 51

If A = [(2, -1, 3),(-4, 5, 1)] and B = [(2, 3),(4, -2),(1, 5)], then ______.

• Only AB is defined

• Only BA is defined

• AB and BA both are defined

• AB and BA both are not defined.

Solved Examples | Q 11 | Page 51

The matrix A = [(0, 0, 5),(0, 5, 0),(5, 0, 0)] is a ______.

• Scalar matrix

• Diagonal matrix

• Unit matrix

• Square matrix

Solved Examples | Q 12 | Page 51

If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.

• Skew symmetric matrix

• Null matrix

• Symmetric matrix

• None of these

#### Fill in the blanks in the Examples 13 to 15

Solved Examples | Q 13 | Page 52

If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.

Solved Examples | Q 14 | Page 52

If A and B are matrices of same order, then (3A –2B)′ is equal to______.

Solved Examples | Q 15 | Page 52

Addition of matrices is defined if order of the matrices is ______.

Solved Examples | Q 16 | Page 52

If two matrices A and B are of the same order, then 2A + B = B + 2A.

• True

• False

Solved Examples | Q 17 | Page 52

Matrix subtraction is associative

• True

• False

Solved Examples | Q 18 | Page 52

For the non singular matrix A, (A′)–1 = (A–1)′.

• True

• False

Solved Examples | Q 19 | Page 52

AB = AC ⇒ B = C for any three matrices of same order.

• True

• False

Exercise [Pages 52 - 64]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 3 MatricesExercise [Pages 52 - 64]

Exercise | Q 1 | Page 52

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Exercise | Q 2. (i) | Page 52

In the matrix A = [("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)], write: The order of the matrix A

Exercise | Q 2. (ii) | Page 52

In the matrix A = [("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)], write: The number of elements

Exercise | Q 2. (iii) | Page 52

In the matrix A = [("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)], write: elements a23, a31, a12

Exercise | Q 3. (i) | Page 53

Construct a2 × 2 matrix where aij = ("i" - 2"j")^2/2

Exercise | Q 3. (ii) | Page 53

Construct a2 × 2 matrix where aij = |–2i + 3j|

Exercise | Q 4 | Page 53

Construct a 3 × 2 matrix whose elements are given by aij = ei.x sinjx.

Exercise | Q 5 | Page 53

Find the values of a and b if A = B, where A = [("a" + 4, 3"b"),(8, -6)], B = [(2"a" + 2, "b"^2 + 2),(8, "b"^2 - 5"b")]

Exercise | Q 6 | Page 53

If possible, find the sum of the matrices A and B, where A = [(sqrt(3), 1),(2, 3)], and B = [(x, y, z),(a, "b", 6)]

Exercise | Q 7. (i) | Page 53

If X = [(3, 1, -1),(5, -2, -3)] and Y = [(2, 1, -1),(7, 2, 4)], find X + Y

Exercise | Q 7. (ii) | Page 53

If X = [(3, 1, -1),(5, -2, -3)] and Y = [(2, 1, -1),(7, 2, 4)], find 2X – 3Y

Exercise | Q 7. (iii) | Page 53

If X = [(3, 1, -1),(5, -2, -3)] and Y = [(2, 1, -1),(7, 2, 4)], find A matrix Z such that X + Y + Z is a zero matrix

Exercise | Q 8 | Page 53

Find non-zero values of x satisfying the matrix equation:

x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]

Exercise | Q 9 | Page 53

If A = [(0, 1),(1, 1)] and B = [(0, -1),(1, 0)], show that (A + B)(A – B) ≠ A2 – B2

Exercise | Q 10 | Page 54

Find the value of x if [(1, x, 1)] [(1, 3, 2),(2, 5,1),(15, 3, 2)] [(1),(2),(x)] = 0

Exercise | Q 11 | Page 54

Show that A = [(5, 3),(-1, -2)] satisfies the equation A2 – 3A – 7I = O and hence find A–1.

Exercise | Q 12 | Page 54

Find the matrix A satisfying the matrix equation:

[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]

Exercise | Q 13 | Page 54

Find A, if [(4),(1),(3)] A = [(-4, 8,4),(-1, 2, 1),(-3, 6, 3)]

Exercise | Q 14 | Page 54

If A = [(3, -4),(1, 1),(2, 0)] and B = [(2, 1, 2),(1, 2, 4)], then verify (BA)2 ≠ B2A2

Exercise | Q 15 | Page 54

If possible, find BA and AB, where A = [(2, 1, 2),(1, 2, 4)], B = [(4, 1),(2, 3),(1, 2)]

Exercise | Q 16 | Page 54

Show by an example that for A ≠ O, B ≠ O, AB = O

Exercise | Q 17 | Page 54

Given A = [(2, 4, 0),(3, 9, 6)] and B = [(1, 4),(2, 8),(1, 3)] is (AB)′ = B′A′?

Exercise | Q 18 | Page 54

Solve for x and y: x[(2),(1)] + y[(3),(5)] + [(-8),(-11)] = O

Exercise | Q 19 | Page 55

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

2X + 3Y = [(2, 3),(4, 0)], 3Y + 2Y = [(-2, 2),(1, -5)]

Exercise | Q 20 | Page 55

If A = [(3, 5)], B = [(7, 3)], then find a non-zero matrix C such that AC = BC.

Exercise | Q 21 | Page 55

Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.

Exercise | Q 22. (i) | Page 55

If A = [(1, 2),(-2, 1)], B = [(2, 3),(3, -4)] and C = [(1, 0),(-1, 0)], verify: (AB)C = A(BC)

Exercise | Q 22. (ii) | Page 55

If A = [(1, 2),(-2, 1)], B = [(2, 3),(3, -4)] and C = [(1, 0),(-1, 0)], verify: A(B + C) = AB + AC

Exercise | Q 23 | Page 55

If P = [(x, 0, 0),(0, y, 0),(0, 0, z)] and Q = [("a", 0, 0),(0, "b", 0),(0, 0, "c")], prove that PQ = [(x"a", 0, 0),(0, y"b", 0),(0, 0, z"c")] = QP

Exercise | Q 24 | Page 55

If: [(2, 1, 3)] [(-1, 0, -1),(-1, 1, 0),(0, 1, 1)] [(1),(0),(-1)] = A, find A

Exercise | Q 25 | Page 55

If A = [(2, 1)], B = [(5, 3, 4),(8, 7, 6)] and C = [(-1, 2, 1),(1, 0, 2)], verify that A(B + C) = (AB + AC).

Exercise | Q 26 | Page 56

If A = [(1, 0, -1),(2, 1, 3 ),(0, 1, 1)], then verify that A2 + A = A(A + I), where I is 3 × 3 unit matrix.

Exercise | Q 27. (i) | Page 56

If A = [(0, -1, 2),(4, 3, -4)] and B = [(4, 0),(1, 3),(2, 6)], then verify that: (A′)′ = A

Exercise | Q 27. (ii) | Page 56

If A = [(0, -1, 2),(4, 3, -4)] and B = [(4, 0),(1, 3),(2, 6)], then verify that: (A′)′ = (AB)' = B'A'

Exercise | Q 27. (iii) | Page 56

If A = [(0, -1, 2),(4, 3, -4)] and B = [(4, 0),(1, 3),(2, 6)], then verify that: (kA)' = (kA')

Exercise | Q 28. (i) | Page 56

If A = [(1, 2),(4, 1),(5, 6)] B = [(1, 2),(6, 4),(7, 3)], then verify that: (2A + B)′ = 2A′ + B′

Exercise | Q 28. (ii) | Page 56

If A = [(1, 2),(4, 1),(5, 6)] B = [(1, 2),(6, 4),(7, 3)], then verify that: (A – B)′ = A′ – B′

Exercise | Q 29 | Page 56

Show that A′A and AA′ are both symmetric matrices for any matrix A.

Exercise | Q 30 | Page 56

Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2? Give reasons.

Exercise | Q 31 | Page 56

Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.

Exercise | Q 32.(a) | Page 56

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: A + (B + C) = (A + B) + C

Exercise | Q 32.(b) | Page 56

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: A(BC) = (AB)C

Exercise | Q 32.(c) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (a + b)B = aB + bB

Exercise | Q 32.(d) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: a(C – A) = aC – aA

Exercise | Q 32.(e) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (AT)T = A

Exercise | Q 32.(f) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (bA)T = bAT

Exercise | Q 32.(g) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (AB)T = BTAT

Exercise | Q 32.(h) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (A – B)C = AC – BC

Exercise | Q 32.(i) | Page 57

Let A = [(1, 2),(-1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, -2)] and a = 4, b = –2. Show that: (A – B)T = AT – BT

Exercise | Q 33 | Page 57

If A = [(costheta, sintheta),(-sintheta, costheta)], then show that A2 = [(cos2theta, sin2theta),(-sin2theta, cos2theta)]

Exercise | Q 34 | Page 57

If A = [(0, -x),(x, 0)], B = [(0, 1),(1, 0)] and x2 = –1, then show that (A + B)2 = A2 + B2

Exercise | Q 35 | Page 57

Verify that A2 = I when A = [(0, 1, -1),(4, -3, 4),(3, -3, 4)]

Exercise | Q 36 | Page 57

Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.

Exercise | Q 37.(i) | Page 57

Find inverse, by elementary row operations (if possible), of the following matrices

[(1, 3),(-5, 7)]

Exercise | Q 37.(ii) | Page 57

Find inverse, by elementary row operations (if possible), of the following matrices

[(1, -3),(-2, 6)]

Exercise | Q 38 | Page 57

If [(xy, 4),(z + 6, x + y)] = [(8, w),(0, 6)], then find values of x, y, z and w.

Exercise | Q 39 | Page 57

If A = [(1, 5),(7, 12)] and B  [(9, 1),(7, 8)], find a matrix C such that 3A + 5B + 2C is a null matrix.

Exercise | Q 40 | Page 58

If A = [(3, -5),(-4, 2)], then find A2 – 5A – 14I. Hence, obtain A3.

Exercise | Q 41 | Page 58

Find the values of a, b, c and d, if 3[("a", "b"),("c", "d")] = [("a", 6),(-1, 2"d")] + [(4, "a" + "b"),("c" + "d", 3)]

Exercise | Q 42 | Page 58

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

Exercise | Q 43 | Page 58

If A = [(1, 2),(4, 1)], find A2 + 2A + 7I.

Exercise | Q 44 | Page 58

If A = [(cosalpha, sinalpha),(-sinalpha, cosalpha)], and A–1 = A′, find value of α

Exercise | Q 45 | Page 58

If the matrix [(0, "a", 3),(2, "b", -1),("c", 1, 0)], is a skew symmetric matrix, find the values of a, b and c.

Exercise | Q 46 | Page 58

If P(x) = [(cosx, sinx),(-sinx, cosx)], then show that P(x) . (y) = P(x + y) = P(y) . P(x)

Exercise | Q 47 | Page 58

If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I..

Exercise | Q 48 | Page 58

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.

Exercise | Q 49 | Page 58

If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn

Exercise | Q 50 | Page 59

Find x, y, z if A = [(0, 2y, z),(x, y, -z),(x, -y, z)] satisfies A′ = A–1.

Exercise | Q 51.(i) | Page 59

If possible, using elementary row transformations, find the inverse of the following matrices

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

Exercise | Q 51.(ii) | Page 59

If possible, using elementary row transformations, find the inverse of the following matrices

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

Exercise | Q 51.(iii) | Page 59

If possible, using elementary row transformations, find the inverse of the following matrices

[(2, 0, -1),(5, 1, 0),(0, 1, 3)]

Exercise | Q 52 | Page 59

Express the matrix [(2, 3, 1),(1, -1, 2),(4, 1, 2)] as the sum of a symmetric and a skew-symmetric matrix.

#### Objective Type Questions from 53 to 67

Exercise | Q 53 | Page 59

The matrix P = [(0, 0, 4),(0, 4, 0),(4, 0, 0)]is a ______.

• Square matrix

• Diagonal matrix

• Unit matrix

• None

Exercise | Q 54 | Page 59

Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is ______.

• 9

• 27

• 81

• 512

Exercise | Q 55 | Page 59

If [(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)], then the value of x + y is ______.

• x = 3, y = 1

• x = 2, y = 3

• x = 2, y = 4

• x = 3, y = 3

Exercise | Q 56 | Page 60

If A = 1/pi [(sin^-1(xpi), tan^-1(x/pi)),(sin^-1(x/pi), cot^-1(pix))], B = 1/pi [(-cos^-1(x/pi), tan^-1 (x/pi)),(sin^-1(x/pi),-tan^-1(pix))], then A – B is equal to ______.

• I

• O

• 2I

• 1/2"I"

Exercise | Q 57 | Page 60

If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is ______.

• m × 3

• 3 × 3

• m × n

• 3 × n

Exercise | Q 58 | Page 60

If A = [(0, 1),(1, 0)], then A2 is equal to ______.

• [(0, 1),(1, 0)]

• [(1, 0),(1, 0)]

• [(0, 1),(0,1)]

• [(1, 0),(0, 1)]

Exercise | Q 59 | Page 60

If matrix A = [aij]2×2, where aij {:(= 1  "if i" ≠ "j"),(= 0  "if i" = "j"):} then A2 is equal to ______.

• I

• A

• 0

• None of these

Exercise | Q 60 | Page 60

The matrix [(1, 0, 0),(0, 2, 0),(0, 0, 4)] is a ______.

• Identity matrix

• Symmetric matrix

• Skew-symmetric matrix

• None of these

Exercise | Q 61 | Page 61

The matrix [(0, -5, 8),(5, 0, 12),(-8, -12, 0)] is a ______.

• Diagonal matrix

• Symmetric matrix

• Skew-symmetric matrix

• Scalar matrix

Exercise | Q 62 | Page 61

If A is matrix of order m × n and B is a matrix such that AB′ and B′A are both defined, then order of matrix B is ______.

• m × m

• n × n

• n × m

• m × n

Exercise | Q 63 | Page 61

If A and B are matrices of same order, then (AB′ – BA′) is a ______.

• Skew-symmetric matrix

• Null matrix

• Symmetric matrix

• Unit matrix

Exercise | Q 64 | Page 61

If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 –7A is equal to ______.

• A

• I – A

• I + A

• 3A

Exercise | Q 65 | Page 61

For any two matrices A and B, we have ______.

• AB = BA

• AB ≠ BA

• AB = O

• None of the above

Exercise | Q 66 | Page 61

On using elementary column operations C2 → C2 – 2C1 in the following matrix equation [(1, -3),(2, 4)] = [(1, -1),(0, 1)] [(3, 1),(2, 4)], we have: ______.

• [(1, -5),(0, 4)] = [(1, -5),(-2, 2)] [(3, -5),(2, 0)]

• [(1, -5),(0, 4)] = [(1, -1),(0, 1)] [(3, -5),(-0, 2)]

• [(1, -5),(2, 0)] = [(1, -3),(0, 1)] [(3, 1),(-2, 4)]

• [(1, -5),(2, 0)] = [(1, -1),(0, 1)] [(3, -5),(2, 0)]

Exercise | Q 67 | Page 62

On using elementary row operation R1 → R1 – 3R2 in the following matrix equation: [(4, 2),(3, 3)] = [(1, 2),(0, 3)] [(2, 0),(1, 1)], we have: ______.

• [(-5, -7),(3, 3)] = [(1, -7),(0, 3)] [(2, 0),(1, 1)]

• [(-5, -7),(3, 3)] = [(1, 2),(0, 3)] [(-1, -3),(1, 1)]

• [(-5, -7),(3, 3)] = [(1, 2),(1, -7)] [(2, 0),(1, 1)]

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

#### Fill in the blanks 68 – 81

Exercise | Q 68 | Page 62

______ matrix is both symmetric and skew-symmetric matrix.

Exercise | Q 69 | Page 62

Sum of two skew-symmetric matrices is always ______ matrix.

Exercise | Q 70 | Page 62

The negative of a matrix is obtained by multiplying it by ______.

Exercise | Q 71 | Page 62

The product of any matrix by the scalar ______ is the null matrix.

Exercise | Q 72 | Page 62

A matrix which is not a square matrix is called a ______ matrix.

Exercise | Q 73 | Page 62

Matrix multiplication is ______ over addition.

Exercise | Q 74 | Page 62

If A is a symmetric matrix, then A3 is a ______  matrix.

Exercise | Q 75 | Page 62

If A is a skew-symmetric matrix, then A2 is a ______.

Exercise | Q 76.(i) | Page 63

If A and B are square matrices of the same order, then (AB)′ = ______.

Exercise | Q 76.(ii) | Page 63

If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)

Exercise | Q 76.(iii) | Page 63

If A and B are square matrices of the same order, then [k (A – B)]′ = ______.

Exercise | Q 77 | Page 63

If A is skew-symmetric, then kA is a ______. (k is any scalar)

Exercise | Q 78.(i) | Page 63

If A and B are symmetric matrices, then AB – BA is a ______.

Exercise | Q 78.(ii) | Page 63

If A and B are symmetric matrices, then BA – 2AB is a ______.

Exercise | Q 79 | Page 63

If A is symmetric matrix, then B′AB is ______.

Exercise | Q 80 | Page 63

If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.

Exercise | Q 81 | Page 63

In applying one or more row operations while finding A–1 by elementary row operations, we obtain all zeros in one or more, then A–1 ______.

#### State whether the following is True or False: 82 to 101

Exercise | Q 82 | Page 63

A matrix denotes a number.

• True

• False

Exercise | Q 83 | Page 63

Matrices of any order can be added.

• True

• False

Exercise | Q 84 | Page 63

Two matrices are equal if they have same number of rows and same number of columns.

• True

• False

Exercise | Q 85 | Page 63

Matrices of different orders can not be subtracted.

• True

• False

Exercise | Q 86 | Page 63

Matrix addition is associative as well as commutative.

• True

• False

Exercise | Q 87 | Page 63

Matrix multiplication is commutative.

• True

• False

Exercise | Q 88 | Page 63

A square matrix where every element is unity is called an identity matrix.

• True

• False

Exercise | Q 89 | Page 63

If A and B are two square matrices of the same order, then A + B = B + A.

• True

• False

Exercise | Q 90 | Page 63

If A and B are two matrices of the same order, then A – B = B – A.

• True

• False

Exercise | Q 91 | Page 63

If matrix AB = O, then A = O or B = O or both A and B are null matrices.

• True

• False

Exercise | Q 92 | Page 63

Transpose of a column matrix is a column matrix.

• True

• False

Exercise | Q 93 | Page 63

If A and B are two square matrices of the same order, then AB = BA.

• True

• False

Exercise | Q 94 | Page 63

If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

• True

• False

Exercise | Q 95 | Page 64

If A and B are any two matrices of the same order, then (AB)′ = A′B′.

• True

• False

Exercise | Q 96 | Page 64

If (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

• True

• False

Exercise | Q 97 | Page 64

If A, B and C are square matrices of same order, then AB = AC always implies that B = C

• True

• False

Exercise | Q 98 | Page 64

AA′ is always a symmetric matrix for any matrix A.

• True

• False

Exercise | Q 99 | Page 64

If A = [(2, 3, -1),(1, 4, 2)] and B = [(2, 3),(4, 5),(2, 1)], then AB and BA are defined and equal.

• True

• False

Exercise | Q 100 | Page 64

If A is skew-symmetric matrix, then A2 is a symmetric matrix.

• True

• False

Exercise | Q 101 | Page 64

(AB)–1 = A–1. B–1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.

• True

• False

## Chapter 3: Matrices

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 3 - Matrices

NCERT solutions for Mathematics Exemplar Class 12 chapter 3 (Matrices) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 12 chapter 3 Matrices are Introduction of Operations on Matrices, Inverse of a Nonsingular Matrix by Elementary Transformation, Multiplication of Two Matrices, Negative of Matrix, Properties of Matrix Addition, Concept of Transpose of a Matrix, Subtraction of Matrices, Addition of Matrices, Symmetric and Skew Symmetric Matrices, Types of Matrices, Proof of the Uniqueness of Inverse, Invertible Matrices, Multiplication of Matrices, Properties of Multiplication of Matrices, Equality of Matrices, Order of a Matrix, Matrices Notation, Introduction of Matrices, Multiplication of a Matrix by a Scalar, Properties of Scalar Multiplication of a Matrix, Properties of Transpose of the Matrices, Elementary Transformations.

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