#### 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

#### NCERT Mathematics Class 12

## Chapter 3 : Matrices

#### Pages 64 - 80

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]`

The order of the matrix

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]`

The number of elements,

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`

Find A - B

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]` Write the elements *a*_{13}, *a*_{21}, *a*_{33}, *a*_{24}, *a*_{23}

If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Construct a 2 × 2 matrix, `A = [a_(ij)]`, whose elements are given by:

`a_(ij) = (i+j)^2/2`

Construct a 2 × 2 matrix, `A= [a_(ij)]`, whose elements are given by `a_(ij) = i/j`

Construct a 2 × 2 matrix, `A = [a_(ij)]` whose elements are given by:

`a_(ij) = (1 + 2j)^2/2`

Construct a 3 × 4 matrix, whose elements are given by `a_(ij) = 1/2 |-3i + j|`

Construct a 3 × 4 matrix, whose elements are given by `a_(ij) = 2i - j`

Find the value of *x*, *y*, and *z* from the following equation:

`[(4,3),(x,5)] = [(y,z),(1,5)]`

Find the value of *x*, *y*, and *z* from the following equation:

`[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]`

Find the value of *x*, *y*, and *z* from the following equation:

`[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]`

Find the value of *a*, *b*, *c*, and *d* from the equation:

`[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]`

`A = [a_(ij)]_(mxxn)` is a square matrix, if

**(A)** *m* < *n*

**(B)** *m* > *n*

**(C)** *m* = *n*

**(D)** None of these

Which of the given values of *x* and *y* make the following pair of matrices equal

`[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]`

**(A)** `x= (-1)/3, y = 7`

**(B)** Not possible to find

**(C)** `y = 7, x = (-2)/3`

**(D)** `x = (-1)/3, y = (-2)/3`

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

**(A)** 27

**(B) **18

**(C)** 81

**(D)** 512

#### Pages 80 - 83

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`

Find A + B

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`

Find 3A -C

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]` Find AB

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`

Find BA

Compute the following: `[(a,b),(-b, a)] + [(a,b),(b,a)]`

Compute the following:

`[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]`

Compute the following:

`[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]`

Compute the following:

`[(cos^2x, sin^2 x),(sin^2 x ,cos^2 x)]+[(sin^2 x, cos^2 x), (cos^2 x, sin^2 x)]`

Compute the indicated products

`[(a,b),(-b,a)][(a,-b),(b,a)]`

Compute the indicated products

`[(1),(2),(3)] [2,3,4]`

Compute the indicated products

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

Compute the indicated products

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

Compute the indicated products

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

Compute the indicated products

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

if `A = [(1,2,-3),(5,0,2),(1,-1,1)], B = [(3,-1,2),(4,2,5),(2,0,3)] and C = [(4,1,2),(0,3,2),(1,-2,3)]` then compute (A + B) and (B - C). Also verify that A + (B -C) = (A +B) - C

if ` A = [(2/3, 1, 5/3), (1/3, 2/3, 4/3),(7/3, 2, 2/3)]` and `B = [(2/5, 3/5,1),(1/5, 2/5, 4/5), (7/5,6/5, 2/5)]` then compute 3A - 5B.

Simplify ` cos theta[(cos theta, sintheta),(-sin theta, cos theta)] + sin theta [(sin theta, -cos theta), (cos theta, sin theta)]`

Find X and Y, if `X + Y = [(7,0),(2,5)] and X - Y = [(3,0),(0,3)]`

Find *X *and *Y*, if `2X + 3Y = [(2,3),(4,0)] and 3X + 2Y = [(2, -1),(-1,5)]`

Find X, if `Y = [(3, 2),(1,4)]` and `2X + Y = [(1, 0),(-3, 2)]`

Find x and y, if `2[(1,3),(0, x)]+[(y,0),(1,2)] = [(5,6),(1,8)]`

Solve the equation for x, y, z and t if `2[(x,z),(y, t)] + 3[(1,-1),(0,2)] = 3[(3,5),(4,6)]`

if `x[2/3] + y[(-1),(1)] = [10/5]`, find values of *x* and *y*.

Given `3[(x,y),(z,w)] = [(x,6),(-1,2W)] + [(4,x+y),(Z+W,3)]` find the values of *x*, *y*, *z* and w

If F(x) = `[(cosx, -sinx,0),(sinx, cosx, 0),(0,0,1)]` show that F(x)F(y) = F(x + y)

Show that `[(5, -1),(6,7)][(2,1),(3,4)] != [(2,1),(3,4)][(5,-1),(6,7)]`

Show that `[(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)]!=[(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]`

Find `A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]`

if `A = [(1,0,2),(0,2,1),(2,0,3)]` , prove that `A^2 - 6A^2 + 7A + 2I = 0`

if A = `[(3, -2),(4,-2)] and l = Matric [(1,0),(0,1)]` find *k* so that `A^2 = kA - 2I`

if `A = [(0, -tan alpha/2), (tan alpha/2, 0)]` and *I* is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

(a) Rs 1,800 (b) Rs 2,000

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Assume *X*, *Y*, *Z*, *W* and *P* are matrices of order 2 x n, 3 x k, 2 x p,n x 3 and respectively. The restriction on *n*, *k* and *p* so that PY + WY will be defined are:

**A.** *k* = 3, *p* = *n*

**B.** *k* is arbitrary, *p* = 2

**C.** *p* is arbitrary, *k* = 3

**D.** *k* = 2, *p* = 3

Assume *X*, *Y*, *Z*, *W* and *P* are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively. If *n* = *p*, then the order of the matrix is &X - 5Z

**A** *p* × 2 **B** 2 × *n* **C** *n* × 3 **D** *p* × *n*

#### Pages 88 - 90

Find the transpose the matrices `[(5),(1/2),(-1)]`

Find the transpose of matrices `[(1,-1),(2,3)]`

Find the transpose of matrices `[(-1,5,6),(sqrt3, 5, 6),(2,3,-1)]`

if `A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that

(A+ B)' = A' + B'

if `A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that

(A- B)' = A' - B'

if `A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))]` then verify that (A + B)' = A' + B'

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

if A' = `[(-2,3),(1,2)] and B = [(-1,0),(1,2)]` then find (A + 2B)'

For the matrices *A* and *B*, verify that (*AB*)′ = B'A' where

`A =[(1),(-4), (3)], B = [-1, 2 1]`

For the matrices *A* and *B*, verify that (*AB*)′ = B'A' where

`A =[(0), (1),(2)] , B =[1 , 5, 7]`

if A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]` then verify that A'A = I

if A = `[(sin alpha, cos alpha),(-cos alpha, sin alpha)]` then verify that A'A = I

Show that the matrix A = `[(1,-1,5),(-1,2,1),(5,1,3)]` is a symmetric matrix

Show that the matrix A = `[(0,1,-1),(-1,0,1),(1,-1,0)]` is a skew symmetric matrix

For the matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix

For the matrix A = `[(1,5),(6,7)]` verify that (A - A') is a skew symmetric matrix

Find `1/2 (A + A') and 1/2 (A -A')` When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`

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

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

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

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

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

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

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

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

If *A*, *B* are symmetric matrices of same order, then *AB* − *BA* is a

**A.** Skew symmetric matrix **B.** Symmetric matrix

**C.** Zero matrix **D. **Identity matrix

if A= `[(cos alpha, -sin alpha),(sin alpha, cos alpha)]` then A + A' = I if the value of α is

A `pi/6`

B `pi/3`

C `pi`

D `(3pi)/2`

#### Pages 97 - 100

Find the inverse of each of the matrices, if it exists. [`(1, -1),(2,3)`]

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`

Find the inverse of each of the matrices, if it exists.` [(2,1),(1,1)]`

if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where *n* is any positive integer

Find the inverse of each of the matrices, if it exists.

`[(1,3),(2,7)]`

Find the inverse of each of the matrices, if it exists.

`[(2,3),(5,7)]`

Find the inverse of each of the matrices, if it exists.

`[(2,7),(1,4)]`

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

[(3,10),(2,7)]

`Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

`[(2,1),(4,2)]`

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Matrices *A* and *B* will be inverse of each other only if

**A.** *AB* = *BA*

**C.** *AB* = 0, *BA* = *I*

**B.** *AB* = *BA* = 0

**D. ***AB* = *BA* = *I*

#### Pages 100 - 101

Let A = `[(0,1),(0,0)]`show that (aI+bA)^{n} = a^{n}I + na^{n-1} bA , where *I* is the identity matrix of order 2 and *n* ∈ **N**

If *A* and *B* are symmetric matrices, prove that *AB* − *BA* is a skew symmetric matrix

Show that the matrix B'AB is symmetric or skew symmetric according as *A* is symmetric or skew symmetric.

Find the values of *x*, *y*, *z* if the matrix `A = [(0,2y,z),(x,y,-z),(x , -y,z)]` satisfy the equation

A'A = I

For what values of x, `[1,2,1] [(1,2,0),(2,0,1),(1,0,2)][(0),(2),(x)]` = O?

if A = `[(3,1),(-1,2)]` show that A^2 - 5A + 7I = O

Find x, if [x, -5, -1][(1,0,2),(0,2,1),(2,0,3)][(x),(4),(1)] = O

A manufacturer produces three products *x*, *y*, *z* which he sells in two markets.

Annual sales are indicated below:

Market |
Products |
||

I |
10000 | 2000 | 18000 |

II |
6000 | 20000 | 8000 |

(a) If unit sale prices of *x*, *y* and *z *are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Find the matrix *X* so that X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`

If *A* and *B* are square matrices of the same order such that *AB* = *BA*, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all *n* ∈ **N**

Choose the correct answer in the following questions:

if A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then

(A) 1 + α² + βγ = 0

(B) 1 – α² + βγ = 0

(C) 1 – α² – βγ = 0

(D) 1 + α² – βγ = 0

If the matrix *A* is both symmetric and skew symmetric, then

**A.** *A* is a diagonal matrix

**B.** *A* is a zero matrix

**C.** *A* is a square matrix

**D. **None of these

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

(A) A

(B) I – A

(C) I

(D) 3A

#### NCERT Mathematics Class 12

#### Textbook solutions for Class 12

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

NCERT solutions for Class 12 Maths 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 Textbook for Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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

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