# NCERT Solutions for Class 12 Maths Chapter 3 Matrices

CBSE Curriculum Syllabus is followed for the preparation of NCERT Solutions for Class 12 Maths Chapter 3 Matrices. This Class 12 maths Ch 3 NCERT Solutions can be consulted by CBSE students & other start board students to help them prepare for their exams. These matrices chapter NCERT solved questions can help you build strong matrices chapter basics.

The Class 12 Maths Matrices NCERT Solutions can also help you improve your problem-solving skills. There are both Hindi and English solutions for class 12 maths chapter 3 matrices. The students can therefore prepare for the exam based on their preferred medium. The NCERT Solutions of Matrix Class 12 Ex 3.1, Ex 3.2, Ex 3.3, Ex 3.4 PDF can also be downloaded from here.

## Matrices Class 12 Ex 3.1

Ex 3.1 Class 12 Maths Question 1.
In the matrix $A=left[ begin{matrix} 2 \ 35 \ sqrt { 3 } end{matrix}begin{matrix} 5 \ -2 \ 1 end{matrix}begin{matrix} 19 \ 5/2 \ -5 end{matrix}begin{matrix} -7 \ 12 \ 17 end{matrix} right]$
(i) The order of the matrix
(ii) The number of elements
(iii) Write the elements a13, a21, a33, a24, a23
Solution:
(i) The matrix A has three rows and 4 columns.
The order of the matrix is 3 x 4.
(ii) There are 3 x 4 = 12 elements in the matrix A
(iii) a13 = 19, a21 = 35, a33 = – 5, a24 = 12, a23 = $\ frac { 5 }{ 2 }$

Ex 3.1 Class 12 Maths Question 2.
If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Solution:
(i) 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6
Thus there are 8 matrices having 24 elements their order are (1 x 24), (24 x 1), (2 x 12), (12 x 2),(3 x 8), (8 x 3), (4 x 6), (6 x 4).
(ii) 13 = 1 x 13,
There are 2 matrices of 13 elements of order (1 x 13) and (13 x 1).

Ex 3.1 Class 12 Maths Question 3.
If a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements.
Solution:
We know that if a matrix is of order m × n, it has mn elements.
=> 18 = 1 x 18 = 2 x 9 = 3 x 6
Thus, all possible ordered pairs of the matrix
having 18 elements are:
(1,18), (18,1), (2,9), (9,2), (3,6), (6,3)
If it has 5 elements, then possible order are: (1,5), (5,1)

Ex 3.1 Class 12 Maths Question 4.
Construct a 2 x 2 matrix, A= [aij] whose elements are given by:
$(i)quad { a }_{ ij }=frac { { (i+j) }^{ 2 } }{ 2 }$
$(ii)quad { a }_{ ij }=frac { i }{ j }$
$(iii)quad { a }_{ ij }=frac { { (i+2j) }^{ 2 } }{ 2 }$
Solution:
$A={ left[ { a }_{ ij } right] }_{ 2times 2 }=begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } \ { a }_{ 21 } & { a }_{ 22 } end{bmatrix}$
$(i)quad { a }_{ ij }=frac { { (i+j) }^{ 2 } }{ 2 }$

Ex 3.1 Class 12 Maths Question 5.
Construct a 3 x 4 matrix , whose elements are given by:
$(i){ a }_{ ij }=frac { 1 }{ 2 } left| -3i+j right|$
$(ii){ a }_{ ij }=2i-j$
Solution:
$A={ left[ { a }_{ ij } right] }_{ 3times 4 }=left[ begin{matrix} { a }_{ 11 } \ { a }_{ 21 } \ { a }_{ 31 } end{matrix}begin{matrix} { a }_{ 12 } \ { a }_{ 22 } \ { a }_{ 32 } end{matrix}begin{matrix} { a }_{ 13 } \ { a }_{ 23 } \ { a }_{ 33 } end{matrix}begin{matrix} { a }_{ 14 } \ { a }_{ 24 } \ { a }_{ 34 } end{matrix} right]$
$(i){ a }_{ ij }=frac { 1 }{ 2 } left| -3i+j right|$

Ex 3.1 Class 12 Maths Question 6.
Find the values of x, y, z from the following equations:
$(i)begin{bmatrix} 4 & 3 \ x & 5 end{bmatrix}=begin{bmatrix} y & z \ 1 & 5 end{bmatrix}$
$(ii)begin{bmatrix} x+y & 2 \ 5+z & xy end{bmatrix}=begin{bmatrix} 6 & 2 \ 5 & 8 end{bmatrix}$
$(iii)left[ begin{matrix} begin{matrix} x+ & y+ & z end{matrix} \ begin{matrix} x & +y end{matrix} \ begin{matrix} y & +z end{matrix} end{matrix} right] =left[ begin{matrix} 9 \ 5 \ 7 end{matrix} right]$
Solution:
$(i)begin{bmatrix} 4 & 3 \ x & 5 end{bmatrix}=begin{bmatrix} y & z \ 1 & 5 end{bmatrix}$
Clearly x = 1,y = 4,z = 3
$(ii)begin{bmatrix} x+y & 2 \ 5+z & xy end{bmatrix}=begin{bmatrix} 6 & 2 \ 5 & 8 end{bmatrix}$
Now 5 + z = 5 => z = 0
Now x + y = 6 and xy = 8
∴ y = 6 – x and x(6 – x) = 8
6x – x² = 8
x² – 6x + 8 = 0
(x – 4)(x – 2) = 0
=>x = 2,4
When x = 2, y = 6 – 2 = 4
and when x = 4,y = 6 – 4 = 2
Hence x = 2,y = 4,z = 0 or x = 4,y = 2,z = 0.
(iii) Equating the corresponding elements.
=> x+y+z=9 …..(i)
x+z = 5 …(ii)
y+ z = 7 …(iii)
x + y + 2z = 12
=> (x+y+z) + z = 12,
9+z = 12 (from equ (i))
z = 3
x + z = 5
=>x + 3 = 5 => x = 2
and y+z = 7
=>y+3 = 7
=> y = 4
=> x = 2, y = 4 and z = 3

Ex 3.1 Class 12 Maths Question 7.
Find the values of a,b,c and d from the equation:
$begin{bmatrix} a-b & 2a+c \ 2a-b & 3c+d end{bmatrix}=begin{bmatrix} -1 & 5 \ 0 & 13 end{bmatrix}$
Solution:
$begin{bmatrix} a-b & 2a+c \ 2a-b & 3c+d end{bmatrix}=begin{bmatrix} -1 & 5 \ 0 & 13 end{bmatrix}$

Ex 3.1 Class 12 Maths Question 8.
A = [aij]m×n is a square matrix, if
(a) m < n (b) n > n
(c) m = n
(d) none of these
Solution:
For a square matrix m=n.
Thus option (c) m = n, is correct.

Ex 3.1 Class 12 Maths Question 9.
Which of the given values of x and y make the following pairs of matrices equal:
$begin{bmatrix} 3x+7 & 5 \ y+1 & 2-3x end{bmatrix},begin{bmatrix} 0 & y-2 \ 8 & 4 end{bmatrix}$
(a) $x=frac { -1 }{ 3 } ,y=7$
(b) Not possible to find
(c) $y=7,x=frac { -2 }{ 3 }$
(d) $x=frac { -1 }{ 3 } ,y=frac { -2 }{ 3 }$
Solution:
$begin{bmatrix} 3x+7 & 5 \ y+1 & 2-3x end{bmatrix},begin{bmatrix} 0 & y-2 \ 8 & 4 end{bmatrix}$
(a) $x=frac { -1 }{ 3 } ,y=7$

Ex 3.1 Class 12 Maths Question 10.
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
Solution:
There are 3 x 3 matrix or 9 entries in matrix each place can be filled with 0 or 1
∴ 9 Places can be filled in 29 = 512 ways
Number of such matrices = 512
Option (d) is correct.

## Matrices Class 12 Ex 3.2

Ex 3.2 Class 12 Maths Question 1.
Let $A=begin{bmatrix} 2 & 4 \ 3 & 2 end{bmatrix},B=begin{bmatrix} 1 & 3 \ -2 & 5 end{bmatrix},C=begin{bmatrix} -2 & 5 \ 3 & 4 end{bmatrix}qquad$
Find each of the following:
(i) A + B
(ii) A – B
(iii) 3A – C
(iv) AB
(v) BA
Solution:
Let $A=begin{bmatrix} 2 & 4 \ 3 & 2 end{bmatrix},B=begin{bmatrix} 1 & 3 \ -2 & 5 end{bmatrix},C=begin{bmatrix} -2 & 5 \ 3 & 4 end{bmatrix}qquad$
(i) A + B

Ex 3.2 Class 12 Maths Question 2.
Compute the following:
$(i)begin{bmatrix} a & quad b \ -b & quad a end{bmatrix}+begin{bmatrix} a & quad b \ b & quad a end{bmatrix}$
$(ii)begin{bmatrix} { a }^{ 2 }+{ b }^{ 2 } & quad { b }^{ 2 }+{ c }^{ 2 } \ { a }^{ 2 }+{ c }^{ 2 } & quad { a }^{ 2 }+{ b }^{ 2 } end{bmatrix}+begin{bmatrix} 2ab & quad 2bc \ -2ac & quad -2ab end{bmatrix}$
$(iii)left[ begin{matrix} begin{matrix} -1 \ 8 \ 2 end{matrix} & begin{matrix} 4 \ 5 \ 8 end{matrix} & begin{matrix} -6 \ 16 \ 5 end{matrix} end{matrix} right] +left[ begin{matrix} begin{matrix} 12 \ 8 \ 3 end{matrix} & begin{matrix} 7 \ 0 \ 2 end{matrix} & begin{matrix} 6 \ 5 \ 4 end{matrix} end{matrix} right]$
$(iv)begin{bmatrix} { cos }^{ 2 }x & quad { sin }^{ 2 }x \ { sin }^{ 2 }x & { quad cos }^{ 2 }x end{bmatrix}+begin{bmatrix} { sin }^{ 2 }x & quad { cos }^{ 2 }x \ { cos }^{ 2 }x & { quad sin }^{ 2 }x end{bmatrix}$
Solution:
$(i)begin{bmatrix} a & quad b \ -b & quad a end{bmatrix}+begin{bmatrix} a & quad b \ b & quad a end{bmatrix}$
$=begin{bmatrix} 2a & quad 2b \ 0 & quad 2a end{bmatrix}$

Ex 3.2 Class 12 Maths Question 3.
Compute the indicated products.
(i) $begin{bmatrix} a & quad b \ -b & quad a end{bmatrix}begin{bmatrix} a & quad -b \ b & quad quad a end{bmatrix}$
(ii) $left[ begin{matrix} 1 \ 2 \ 3 end{matrix} right] left[ begin{matrix} 2 & 3 & 4 end{matrix} right]$
(iii) $begin{bmatrix} 1 & -2 \ 2 & quad 3 end{bmatrix}left[ begin{matrix} 1 & 2 & 3 \ 2 & 3 & 1 end{matrix} right]$
(iv) $left[ begin{matrix} 2 & 3 & 4 \ 3 & 4 & 5 \ 4 & 5 & 6 end{matrix} right] left[ begin{matrix} 1 & -3 & 5 \ 0 & 2 & 4 \ 3 & 0 & 5 end{matrix} right]$
(v) $left[ begin{matrix} 2 \ 3 \ -1 end{matrix}begin{matrix} 1 \ 2 \ 1 end{matrix} right] left[ begin{matrix} begin{matrix} 1 & 0 & 1 end{matrix} \ begin{matrix} -1 & 2 & 1 end{matrix} end{matrix} right]$
(vi) $left[ begin{matrix} begin{matrix} 3 & -1 & 3 end{matrix} \ begin{matrix} -1 & 0 & 2 end{matrix} end{matrix} right] left[ begin{matrix} begin{matrix} 2 \ 1 \ 3 end{matrix} & begin{matrix} -3 \ 0 \ 1 end{matrix} end{matrix} right]$
Solution:
(i) $begin{bmatrix} a & quad b \ -b & quad a end{bmatrix}begin{bmatrix} a & quad -b \ b & quad quad a end{bmatrix}$
$begin{bmatrix} { a }^{ 2 }+{ b }^{ 2 } & 0 \ 0 & { b }^{ 2 }+{ a }^{ 2 } end{bmatrix}$

Ex 3.2 Class 12 Maths Question 4.
If $A=left[ begin{matrix} 1 & 2 & -3 \ 5 & 0 & 2 \ 1 & -1 & 1 end{matrix} right] ,B=left[ begin{matrix} 3 & -1 & 2 \ 4 & 2 & 5 \ 2 & 0 & 3 end{matrix} right] ,C=left[ begin{matrix} 4 & 1 & 2 \ 0 & 3 & 2 \ 1 & -2 & 3 end{matrix} right]$
then compute (A + B) and (B – C). Also verify that A + (B – C) = (A + B) – C.
Solution:
Given
$A=left[ begin{matrix} 1 & 2 & -3 \ 5 & 0 & 2 \ 1 & -1 & 1 end{matrix} right] ,B=left[ begin{matrix} 3 & -1 & 2 \ 4 & 2 & 5 \ 2 & 0 & 3 end{matrix} right] ,C=left[ begin{matrix} 4 & 1 & 2 \ 0 & 3 & 2 \ 1 & -2 & 3 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 5.
If $A=left[ begin{matrix} frac { 2 }{ 3 } & 1 & frac { 5 }{ 3 } \ frac { 1 }{ 3 } & frac { 2 }{ 3 } & frac { 4 }{ 3 } \ frac { 7 }{ 3 } & 2 & frac { 2 }{ 3 } end{matrix} right] andquad B=left[ begin{matrix} frac { 2 }{ 5 } & frac { 3 }{ 5 } & 1 \ frac { 1 }{ 5 } & frac { 2 }{ 5 } & frac { 4 }{ 5 } \ frac { 7 }{ 5 } & frac { 6 }{ 5 } & frac { 2 }{ 5 } end{matrix} right] ,$
then compute 3A – 5B.
Solution:
$3A-5B=3left[ begin{matrix} frac { 2 }{ 3 } & 1 & frac { 5 }{ 3 } \ frac { 1 }{ 3 } & frac { 2 }{ 3 } & frac { 4 }{ 3 } \ frac { 7 }{ 3 } & 2 & frac { 2 }{ 3 } end{matrix} right] -5left[ begin{matrix} frac { 2 }{ 5 } & frac { 3 }{ 5 } & 1 \ frac { 1 }{ 5 } & frac { 2 }{ 5 } & frac { 4 }{ 5 } \ frac { 7 }{ 5 } & frac { 6 }{ 5 } & frac { 2 }{ 5 } end{matrix} right] ,$
$left[ begin{matrix} 2 & 3 & 5 \ 1 & 2 & 4 \ 7 & 6 & 2 end{matrix} right] -left[ begin{matrix} 2 & 3 & 5 \ 1 & 2 & 4 \ 7 & 6 & 2 end{matrix} right] =left[ begin{matrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 6.
Simplify:
$costheta begin{bmatrix} costheta & sintheta \ -sintheta & costheta end{bmatrix}+sintheta begin{bmatrix} sintheta & -costheta \ costheta & sintheta end{bmatrix}$
Solution:
$costheta begin{bmatrix} costheta & sintheta \ -sintheta & costheta end{bmatrix}+sintheta begin{bmatrix} sintheta & -costheta \ costheta & sintheta end{bmatrix}$

Ex 3.2 Class 12 Maths Question 7.
Find X and Y if
$(i)quad X+Y=begin{bmatrix} 7 & 0 \ 2 & 5 end{bmatrix}andquad X-Y=begin{bmatrix} 3 & 0 \ 0 & 3 end{bmatrix}$
$(ii)quad 2X+3Y=begin{bmatrix} 2 & 0 \ 4 & 0 end{bmatrix}andquad 3X+2Y=begin{bmatrix} 2 & -2 \ -1 & 5 end{bmatrix}$
Solution:
$(i)quad X+Y=begin{bmatrix} 7 & 0 \ 2 & 5 end{bmatrix}andquad X-Y=begin{bmatrix} 3 & 0 \ 0 & 3 end{bmatrix}$

Ex 3.2 Class 12 Maths Question 8.
Find
$Xquad ifquad Y=begin{bmatrix} 3 & 2 \ 1 & 4 end{bmatrix}andquad 2X+Y=begin{bmatrix} 1 & 0 \ -3 & 2 end{bmatrix}$
Solution:
$Y=begin{bmatrix} 3 & 2 \ 1 & 4 end{bmatrix}$
We are given that

Ex 3.2 Class 12 Maths Question 9.
Find x and y, if $2begin{bmatrix} 1 & 3 \ 0 & x end{bmatrix}+begin{bmatrix} y & 0 \ 1 & 2 end{bmatrix}=begin{bmatrix} 5 & 6 \ 1 & 8 end{bmatrix}$
Solution:
$2begin{bmatrix} 1 & 3 \ 0 & x end{bmatrix}+begin{bmatrix} y & 0 \ 1 & 2 end{bmatrix}=begin{bmatrix} 5 & 6 \ 1 & 8 end{bmatrix}$
=> $begin{bmatrix} 2+y & quad 6 \ 1 & quad 2x+2 end{bmatrix}=begin{bmatrix} 5 & 6 \ 1 & 8 end{bmatrix}$
=> 2+y = 5 and 2x+2 = 8
=> y=3 and x=3
Hence x=3 and y=3

Ex 3.2 Class 12 Maths Question 10.
Solve the equation for x,y,z and t, if
$2begin{bmatrix} x & z \ y & t end{bmatrix}+3begin{bmatrix} 1 & -1 \ 0 & 2 end{bmatrix}=3begin{bmatrix} 3 & 5 \ 4 & 6 end{bmatrix}$
Solution:
$2begin{bmatrix} x & z \ y & t end{bmatrix}+3begin{bmatrix} 1 & -1 \ 0 & 2 end{bmatrix}=3begin{bmatrix} 3 & 5 \ 4 & 6 end{bmatrix}$

Ex 3.2 Class 12 Maths Question 11.
If $xleft[ begin{matrix} 2 \ 3 end{matrix} right] +yleft[ begin{matrix} -1 \ 1 end{matrix} right] =left[ begin{matrix} 10 \ 5 end{matrix} right]$ then find the values of x and y
Solution:
$xleft[ begin{matrix} 2 \ 3 end{matrix} right] +yleft[ begin{matrix} -1 \ 1 end{matrix} right] =left[ begin{matrix} 10 \ 5 end{matrix} right]$
=> $left[ begin{matrix} 2x-y \ 3x+y end{matrix} right] =left[ begin{matrix} 10 \ 5 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 12.
Given
$3begin{bmatrix} x & quad y \ z & quad w end{bmatrix}=begin{bmatrix} x & quad 6 \ -1 & quad 2w end{bmatrix}+begin{bmatrix} 4 & quad x+y \ z+w & 3 end{bmatrix}$
find the values of x,y,z and w.
Solution:
$3begin{bmatrix} x & quad y \ z & quad w end{bmatrix}=begin{bmatrix} x & quad 6 \ -1 & quad 2w end{bmatrix}+begin{bmatrix} 4 & quad x+y \ z+w & 3 end{bmatrix}$
=> $begin{bmatrix} 3x & quad 3y \ 3z & quad 3w end{bmatrix}=begin{bmatrix} x+4 & quad 6+x+y \ -1+z+w & quad 2w+3 end{bmatrix}$
=> 3x = x + 4 => x = 2
and 3y = 6 + x + y => y = 4
Also, 3w = 2w + 3 => w = 3
Again, 3z = – 1 + z + w
=> 2z = – 1 + 3
=> 2z = 2
=> z = 1
Hence x = 2 ,y = 4, z = 1, w = 3.

Ex 3.2 Class 12 Maths Question 13.
If F(x) = $left[ begin{matrix} cosx & -sinx & 0 \ sinx & cosx & 0 \ 0 & 0 & 1 end{matrix} right]$
then show that F(x).F(y) = F(x+y)
Solution:
F(x) = $left[ begin{matrix} cosx & -sinx & 0 \ sinx & cosx & 0 \ 0 & 0 & 1 end{matrix} right]$
∴ F(y) = $left[ begin{matrix} cosy & -siny & 0 \ siny & cosy & 0 \ 0 & 0 & 1 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 14.
Show that
$(i)begin{bmatrix} 5 & -1 \ 6 & 7 end{bmatrix}begin{bmatrix} 2 & 1 \ 3 & 4 end{bmatrix}neq begin{bmatrix} 2 & 1 \ 3 & 4 end{bmatrix}begin{bmatrix} 5 & -1 \ 6 & 7 end{bmatrix}$
$(ii)left[ begin{matrix} 1 & 2 & 3 \ 0 & 1 & 0 \ 1 & 1 & 0 end{matrix} right] left[ begin{matrix} -1 & 1 & 0 \ 0 & -1 & 1 \ 2 & 3 & 4 end{matrix} right] neq left[ begin{matrix} -1 & 1 & 0 \ 0 & -1 & 1 \ 2 & 3 & 4 end{matrix} right] left[ begin{matrix} 1 & 2 & 3 \ 0 & 1 & 0 \ 1 & 1 & 0 end{matrix} right]$
Solution:
$(i)L.H.S=begin{bmatrix} 5 & -1 \ 6 & 7 end{bmatrix}begin{bmatrix} 2 & 1 \ 3 & 4 end{bmatrix}=begin{bmatrix} 7 & 1 \ 33 & 34 end{bmatrix}$
$R.H.S=begin{bmatrix} 2 & 1 \ 3 & 4 end{bmatrix}begin{bmatrix} 5 & -1 \ 6 & 7 end{bmatrix}=begin{bmatrix} 16 & 5 \ 39 & 25 end{bmatrix}$
L.H.S≠R.H.S

Ex 3.2 Class 12 Maths Question 15.
Find A² – 5A + 6I, if A = $left[ begin{matrix} 2 & 0 & 1 \ 2 & 1 & 3 \ 1 & -1 & 0 end{matrix} right]$
Solution:
A² – 5A + 6I = $left[ begin{matrix} 2 & 0 & 1 \ 2 & 1 & 3 \ 1 & -1 & 0 end{matrix} right] left[ begin{matrix} 2 & 0 & 1 \ 2 & 1 & 3 \ 1 & -1 & 0 end{matrix} right] -5left[ begin{matrix} 2 & 0 & 1 \ 2 & 1 & 3 \ 1 & -1 & 0 end{matrix} right] +6left[ begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 16.
If A = $left[ begin{matrix} 1 & 0 & 2 \ 0 & 2 & 1 \ 2 & 0 & 3 end{matrix} right]$ Prove that A³-6A²+7A+2I = 0
Solution:
We have
A² = A x A
$left[ begin{matrix} 1 & 0 & 2 \ 0 & 2 & 1 \ 2 & 0 & 3 end{matrix} right] times left[ begin{matrix} 1 & 0 & 2 \ 0 & 2 & 1 \ 2 & 0 & 3 end{matrix} right] =left[ begin{matrix} 5 & 0 & 8 \ 2 & 4 & 5 \ 8 & 0 & 13 end{matrix} right]$

Ex 3.2 Class 12 Maths Question 17.
If $A=begin{bmatrix} 3 & -2 \ 4 & -2 end{bmatrix},I=begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix}$ then find k so that A²=kA-2I
Solution:
Given
$A=begin{bmatrix} 3 & -2 \ 4 & -2 end{bmatrix},I=begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix}$
Required: To find the value of k
Now A²=kA-2I

Ex 3.2 Class 12 Maths Question 18.
If $A=begin{bmatrix} 0 & -tanfrac { alpha }{ 2 } \ tanfrac { alpha }{ 2 } & 0 end{bmatrix}$ and I is the identity matrix of order 2,show that
$I+A=I-Abegin{bmatrix} cosalpha & quad -sinalpha \ sinalpha & quad cosalpha end{bmatrix}$
Solution:
L.H.S=$I+A=begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix}+begin{bmatrix} 0 & -tanfrac { alpha }{ 2 } \ tanfrac { alpha }{ 2 } & 0 end{bmatrix}$

Ex 3.2 Class 12 Maths Question 19.
A trust has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year and second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bond if the trust fund obtains an annual total interest of
(a) Rs 1800
(b) Rs 2000
Solution:
Let Rs 30,000 be divided into two parts and Rs x and Rs (30,000-x)
Let it be represented by 1 x 2 matrix [x (30,000-x)]
Rate of interest is 005 and 007 per rupee.
It is denoted by the matrix R of order 2 x 1.

Ex 3.2 Class 12 Maths Question 20.
The book-shop of a particular school has 10 dozen Chemistry books, 8 dozen Physics books, 10 dozen Economics books. Their selling price are Rs 80, Rs 60 and Rs 40 each respectively. Find die total amount the book-shop will receive from selling all the books using matrix algebra.
Solution:
Number of Chemistry books = 10 dozen books
= 120 books
Number of Physics books = 8 dozen books = 96 books
Number of Economics books = 10 dozen books
= 120 books

Assuming X, Y, Z, W and P are the matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k respectively. Choose the correct answer in Question 21 and 22.

Ex 3.2 Class 12 Maths Question 21.
The restrictions 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) pis arbitrary, k = 3
(d) k = 2,p = 3
Solution:
Given : x2xn, y3xn, z2xp, wnx3, Ppxk
Now py +wy = Ppxk x y3+k x wnx3  x y3xk
Clearly, k = 3 and p = n
Hence, option (a) is correct p x 2.

Ex 3.2 Class 12 Maths Question 22.
If n = p, then the order of the matrix 7X – 5Z is:
(a) p x 2
(b) 2 x n
(c) n x 3
(d) p x n.
Solution:
7X – 5Z = 7X2xn – 5X2xp
∴ We can add two matrices if their order is same n = P
∴ Order of 7X – 5Z is 2 x n.
Hence, option (b) is correct 2 x n.

## Matrices Class 12 Ex 3.3

Ex 3.3 Class 12 Maths Question 1.
Find the transpose of each of the following matrices:
(i) $left[ begin{matrix} 5 \ frac { 1 }{ 2 } \ -1 end{matrix} right]$
(ii) $begin{bmatrix} 1 & -1 \ 2 & 3 end{bmatrix}$
(iii) $left[ begin{matrix} -1 & 5 & 6 \ sqrt { 3 } & 5 & 6 \ 2 & 3 & -1 end{matrix} right]$
Solution:
(i) let A = $left[ begin{matrix} 5 \ frac { 1 }{ 2 } \ -1 end{matrix} right]$
∴ transpose of A = A’ = $left[ begin{matrix} 5 & frac { 1 }{ 2 } & -1 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 2.
If $A=left[ begin{matrix} -1 & 2 & 3 \ 5 & 7 & 9 \ -2 & 1 & 1 end{matrix} right] ,B=left[ begin{matrix} -4 & 1 & -5 \ 1 & 2 & 0 \ 1 & 3 & 1 end{matrix} right]$
then verify that:
(i) (A+B)’=A’+B’
(ii) (A-B)’=A’-B’
Solution:
$A=left[ begin{matrix} -1 & 2 & 3 \ 5 & 7 & 9 \ -2 & 1 & 1 end{matrix} right] ,B=left[ begin{matrix} -4 & 1 & -5 \ 1 & 2 & 0 \ 1 & 3 & 1 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 3.
If $A'=left[ begin{matrix} 3 & 4 \ -1 & 2 \ 0 & 1 end{matrix} right] ,B=left[ begin{matrix} -1 & 2 & 1 \ 1 & 2 & 3 end{matrix} right]$
then verify that:
(i) (A+B)’ = A’+B’
(ii) (A-B)’ = A’-B’
Solution:
$A'=left[ begin{matrix} 3 & 4 \ -1 & 2 \ 0 & 1 end{matrix} right] ,B=left[ begin{matrix} -1 & 2 & 1 \ 1 & 2 & 3 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 4.
If $A'=begin{bmatrix} -2 & 3 \ 1 & 2 end{bmatrix},B=begin{bmatrix} -1 & 0 \ 1 & 2 end{bmatrix}$
then find (A+2B)’
Solution:
$A'=begin{bmatrix} -2 & 3 \ 1 & 2 end{bmatrix},B=begin{bmatrix} -1 & 0 \ 1 & 2 end{bmatrix}$

Ex 3.3 Class 12 Maths Question 5.
For the matrices A and B, verify that (AB)’ = B’A’, where
$(i)quad A=left[ begin{matrix} 1 \ -4 \ 3 end{matrix} right] ,B=left[ begin{matrix} -1 & 2 & 1 end{matrix} right]$
$(ii)quad A=left[ begin{matrix} 0 \ 1 \ 2 end{matrix} right] ,B=left[ begin{matrix} 1 & 5 & 7 end{matrix} right]$
Solution:
$(i)quad A=left[ begin{matrix} 1 \ -4 \ 3 end{matrix} right]$
$A'=left[ begin{matrix} 1 & -4 & 3 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 6.
If (i) $A=begin{bmatrix} cosalpha & quad sinalpha \ -sinalpha & quad cosalpha end{bmatrix}$ ,the verify that A’A=I
If (ii) $A=begin{bmatrix} sinalpha & quad cosalpha \ -cosalpha & quad sinalpha end{bmatrix}$,the verify that A’A=I
Solution:
(i) $A=begin{bmatrix} sinalpha & quad cosalpha \ -sinalpha & quad cosalpha end{bmatrix}$
$A'=begin{bmatrix} cosalpha & quad -sinalpha \ sinalpha & quad cosalpha end{bmatrix}$

Ex 3.3 Class 12 Maths Question 7.
(i) Show that the matrix $A=left[ begin{matrix} 1 & -1 & 5 \ -1 & 2 & 1 \ 5 & 1 & 3 end{matrix} right]$ is a symmetric matrix.
(ii) Show that the matrix $A=left[ begin{matrix} 0 & 1 & -1 \ -1 & 0 & 1 \ 1 & -1 & 0 end{matrix} right]$ is a skew-symmetric matrix.
Solution:
(i) For a symmetric matrix aij = aji
Now,
$A=left[ begin{matrix} 1 & -1 & 5 \ -1 & 2 & 1 \ 5 & 1 & 3 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 8.
For the matrix, $A=begin{bmatrix} 1 & 5 \ 6 & 7 end{bmatrix}$
(i) (A+A’) is a symmetric matrix.
(ii) (A-A’) is a skew-symmetric matrix.
Solution:
$A=begin{bmatrix} 1 & 5 \ 6 & 7 end{bmatrix}$
=> $A'=begin{bmatrix} 1 & 6 \ 5 & 7 end{bmatrix}$

Ex 3.3 Class 12 Maths Question 9.
Find $\ frac { 1 }{ 2 } (A+A')$ and $\ frac { 1 }{ 2 } (A-A')$,when
$A=left[ begin{matrix} 0 & a & b \ -a & 0 & c \ -b & -c & 0 end{matrix} right]$
Solution:
$A=left[ begin{matrix} 0 & a & b \ -a & 0 & c \ -b & -c & 0 end{matrix} right]$
$A'=left[ begin{matrix} 0 & -a & -b \ a & 0 & -c \ b & c & 0 end{matrix} right]$

Ex 3.3 Class 12 Maths Question 10.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix.
(i)$begin{bmatrix} 3 & 5 \ 1 & -1 end{bmatrix}$
(ii)$left[ begin{matrix} 6 & -2 & 2 \ -2 & 3 & -1 \ 2 & -1 & 3 end{matrix} right]$
(iii)$left[ begin{matrix} 3 & 3 & -1 \ -2 & -2 & 1 \ -4 & -5 & 2 end{matrix} right]$
(iv)$begin{bmatrix} 1 & 5 \ -1 & 2 end{bmatrix}$
Solution:
(i) let $A=begin{bmatrix} 3 & 5 \ 1 & -1 end{bmatrix}$
=> $A'=begin{bmatrix} 3 & 1 \ 5 & -1 end{bmatrix}$

Ex 3.3 Class 12 Maths Question 11.
Choose the correct answer in the following questions:
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
Solution:
Now A’ = B, B’ = B
(AB-BA)’ = (AB)’-(BA)’
= B’A’ – A’B’
= BA-AB
= – (AB – BA)
AB – BA is a skew-symmetric matrix Hence, option (a) is correct.

Ex 3.3 Class 12 Maths Question 12.
If $A=begin{bmatrix} cosalpha & quad -sinalpha \ sinalpha & quad cosalpha end{bmatrix}$ then A+A’ = I, if the
value of α is
(a) $frac { pi }{ 6 }$
(b) $frac { pi }{ 3 }$
(c) π
(d) $frac { 3pi }{ 2 }$
Solution:
Now

Thus option (b) is correct.

## Matrices Class 12 Ex 3.4

Ex 3.4 Class 12 Maths Question 1.
$begin{bmatrix} 1 & -1 \ 2 & 3 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 1 & -1 \ 2 & 3 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 2.
$begin{bmatrix} 2 & 1 \ 1 & 1 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & 1 \ 1 & 1 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 3.
$begin{bmatrix} 1 & 3 \ 2 & 7 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 1 & 3 \ 2 & 7 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 4.
$begin{bmatrix} 2 & 3 \ 5 & 7 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & 3 \ 5 & 7 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 5.
$begin{bmatrix} 2 & 1 \ 7 & 4 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & 1 \ 7 & 4 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 6.
$begin{bmatrix} 2 & 5 \ 1 & 3 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & 5 \ 1 & 3 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 7.
$begin{bmatrix} 3 & 1 \ 5 & 2 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 3 & 1 \ 5 & 2 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 8.
$begin{bmatrix} 4 & 5 \ 3 & 4 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 4 & 5 \ 3 & 4 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 9.
$begin{bmatrix} 3 & 10 \ 2 & 7 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 3 & 10 \ 2 & 7 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Ex 3.4 Class 12 Maths Question 10.
$begin{bmatrix} 3 & -1 \ -4 & 2 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 3 & -1 \ -4 & 2 end{bmatrix}$

Ex 3.4 Class 12 Maths Question 11.
$begin{bmatrix} 2 & -6 \ 1 & -2 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & -6 \ 1 & -2 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 12.
$begin{bmatrix} 6 & -3 \ -2 & 1 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 6 & -3 \ -2 & 1 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 13.
$begin{bmatrix} 2 & -3 \ -1 & 2 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & -3 \ -1 & 2 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 14.
$begin{bmatrix} 2 & 1 \ 4 & 2 end{bmatrix}$
Solution:
Let $A=begin{bmatrix} 2 & 1 \ 4 & 2 end{bmatrix}$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 15.
$left[ begin{matrix} 2 & -3 & 3 \ 2 & 2 & 3 \ 3 & -2 & 2 end{matrix} right]$
Solution:
Let $A=left[ begin{matrix} 2 & -3 & 3 \ 2 & 2 & 3 \ 3 & -2 & 2 end{matrix} right]$

Ex 3.4 Class 12 Maths Question 16.
$left[ begin{matrix} 1 & 3 & -2 \ -3 & 0 & -5 \ 2 & 5 & 2 end{matrix} right]$
Solution:
Let $A=left[ begin{matrix} 1 & 3 & -2 \ -3 & 0 & -5 \ 2 & 5 & 2 end{matrix} right]$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 17.
$left[ begin{matrix} 2 & 0 & -1 \ 5 & 1 & 0 \ 0 & 1 & 3 end{matrix} right]$
Solution:
Let $A=left[ begin{matrix} 2 & 0 & -1 \ 5 & 1 & 0 \ 0 & 1 & 3 end{matrix} right]$
We know that
A = IA

Ex 3.4 Class 12 Maths Question 18.
Choose the correct answer in the following question:
Matrices A and B will be inverse of each other only if
(a) AB = BA
(b) AB = BA = 0
(c) AB = 0,BA = 1
(d) AB = BA = I
Solution:
Choice (d) is correct
i.e., AB = BA = I