#### Online Mock Tests

#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity And Differentiability

Chapter 6: Application Of Derivatives

Chapter 7: Integrals

Chapter 8: Application Of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## Chapter 3: Matrices

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

#### Short Answer

Construct a matrix A = [a_{ij}]_{2×2} whose elements a_{ij} are given by a_{ij} = e^{2ix} sin jx.

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?

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

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

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`

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)]`

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

Let A = `[(2, 3),(-1, 2)]`. Then show that A^{2} – 4A + 7I = O. Using this result calculate A^{5} also.

#### Objective Type Questions Examples 9 to 12

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

A

^{2}– B^{2}A

^{2}– BA – AB – B^{2}A

^{2}– B^{2}+ BA – ABA

^{2}– BA + B^{2}+ AB

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.

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

Scalar matrix

Diagonal matrix

Unit matrix

Square matrix

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

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

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

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

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

True

False

Matrix subtraction is associative

True

False

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

True

False

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

True

False

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

#### Short Answer

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

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

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

In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: elements a_{23}, a_{31}, a_{12}

Construct a_{2 × 2} matrix where a_{ij} = `("i" - 2"j")^2/2`

Construct a_{2 × 2} matrix where a_{ij} = |–2i + 3j|

Construct a 3 × 2 matrix whose elements are given by a_{ij} = e^{i.x} sinjx.

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")]`

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)]`

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

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

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

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)]`

If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A^{2} – B^{2}

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

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

Find the matrix A satisfying the matrix equation:

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

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

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

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

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

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

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

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)]`

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

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

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

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

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

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

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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} = bA^{T}

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} = B^{T}A^{T}

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

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} = A^{T} – B^{T}

If A = `[(costheta, sintheta),(-sintheta, costheta)]`, then show that A^{2} = `[(cos2theta, sin2theta),(-sin2theta, cos2theta)]`

If A = `[(0, -x),(x, 0)]`, B = `[(0, 1),(1, 0)]` and x^{2} = –1, then show that (A + B)^{2} = A^{2} + B^{2}.

Verify that A^{2} = I when A = `[(0, 1, -1),(4, -3, 4),(3, -3, 4)]`

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

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

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

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

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

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

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.

If A = `[(3, -5),(-4, 2)]`, then find A^{2} – 5A – 14I. Hence, obtain A^{3}.

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)]`

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

If A = `[(1, 2),(4, 1)]`, find A^{2} + 2A + 7I.

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

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.

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

If A is square matrix such that A^{2} = A, show that (I + A)^{3} = 7A + I..

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

#### Long Answer

If AB = BA for any two square matrices, prove by mathematical induction that (AB)^{n} = A^{n}B^{n}

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

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

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

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

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

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

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

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

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

Square matrix

Diagonal matrix

Unit matrix

None

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

9

27

81

512

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

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"`

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

If A = `[(0, 1),(1, 0)]`, then A^{2} is equal to ______.

`[(0, 1),(1, 0)]`

`[(1, 0),(1, 0)]`

`[(0, 1),(0,1)]`

`[(1, 0),(0, 1)]`

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

I

A

0

None of these

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

Identity matrix

Symmetric matrix

Skew-symmetric matrix

None of these

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

Diagonal matrix

Symmetric matrix

Skew-symmetric matrix

Scalar matrix

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

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

Skew-symmetric matrix

Null matrix

Symmetric matrix

Unit matrix

If A is a square matrix such that A^{2} = I, then (A – I)^{3} + (A + I)^{3} –7A is equal to ______.

A

I – A

I + A

3A

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

AB = BA

AB ≠ BA

AB = O

None of the above

On using elementary column operations C_{2} → C_{2} – 2C_{1} 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)]`

On using elementary row operation R_{1} → R_{1} – 3R_{2} 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

______ matrix is both symmetric and skew-symmetric matrix.

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

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

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

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

Matrix multiplication is ______ over addition.

If A is a symmetric matrix, then A^{3} is a ______ matrix.

If A is a skew-symmetric matrix, then A^{2} is a ______.

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

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

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

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

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

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

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

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

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

A matrix denotes a number.

True

False

Matrices of any order can be added.

True

False

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

True

False

Matrices of different orders can not be subtracted.

True

False

Matrix addition is associative as well as commutative.

True

False

Matrix multiplication is commutative.

True

False

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

True

False

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

True

False

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

True

False

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

True

False

Transpose of a column matrix is a column matrix.

True

False

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

True

False

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

True

False

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

True

False

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

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

True

False

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

True

False

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

If A is skew-symmetric matrix, then A^{2} is a symmetric matrix.

True

False

(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

## 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.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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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