Download CHAPTER – 1 MATRICES AND DETERMINANTS Points to

CHAPTER – 1
MATRICES AND DETERMINANTS
Points to remember
1.
2.
A matrix obtained by changing rows into columns and columns into rows is called transpose of
given matrix. If the matrix is denoted by a, its transpose is denoted by a ’ or At.
A square matrix A = [aij] is symmetric if A’ = A.
3.
A square matrix A = [aij] is skew symmetric if A’ = –A.
4.
Diagonal elements of a skew symmetric matrix are always zero.
5.
If a determinant corresponding to a square matrix is zero, then matrix is said to be singular matrix.
If |A|  0, then is its said to be a non-singular matrix.
For a square matrix A, A+ A’ is a symmetric matrix and A – A’ is a skew symmetric matrix and
1
1
A   A  A'    A – A' 
2
2
6.
SQUARE MATRIX = SYMMETRIX + SKEW-SYMMETRIC.
7.
(b)
(c)
(d)
8.
(a)
Matrix multiplication is not commutative.
Matrix multiplication is associative, if A, B, C be three matrices of order mxn, nxp and pxq. Then
(AB)C = A(BC)
Matrix multiplication is distributive over addition. If A, B, C be three matrices of order mxn, nxp,
nxp respectively, then A(B + C) = AB + AC.
AI = A = IA
An = A A …………. A (n factors)
Properties of determinants
(i)
(ii)
Value of determinant remains unchanged if its rows and columns are interchanged.
If any two rows or columns are interchanged the value of determinant changes by minus sign.
(iii) If any two rows or columns of a determinant are identical, then its value is zero.
(iv) If each element of a row or column is multiplied by a constant k then the value of determinant
becomes k times the value of original determinant.
(v)
If each element of a row or column is expressed as a sum of two or more terms then the determinant
can be expressed as the sum of two or more determinants.
a1  1
a2   2
a3  3
b1
b2
c1
c2
Graphics By:- Pradeep
a1
a2
a3
1
2
3
b3
 b1
b2
b3  b1
b2
b3
c3
c1
c2
c3
c2
c3
1
c1
Written By:- Raj Kumar Badhan
(vi) If each element of a row or column of a determinant be multiplied by the same constant and then
added to the corresponding elements of some other row or column then the value of determinant
remains unchanged.
(vii) If each element of a row or column is zero then the value of the determinant is zero.
Evaluation of Determinant:
Steps: (i) Reduce large numerals into smaller by applying properties.
(ii) Try to introduce zero at maximum number of places in a particular row or column using properties
and expand along that particular row.
9
9
12
Example : Evaluate,   1 3 4
1 9 12
Take 3 common from C2 & 4 from C3 then
9
3
3
  3  4 1 1 1  3  4  0  0  C2 and C3 are identical 
1
3
Area of a triangle with vertices  x1 , y1  ,  x2 , y2  and  x3 , y3  is
(a)
x1
1
x2
2
x3
y1 1
y2 1  0
y3 1
If points  x1 , y1  ,  x2 , y2  and  x3 , y3  are collinear then
(b)
x1
1
x2
2
x3
10.
3
y1 1
y2 1  0
y3 1
Solution of system of equations using determinants (CRAMER’S RULE):Suppose system of equations is
a1 x  b1 y  c1 z  d1
a2 x  b2 y  c2 z  d 2
a3 x  b3 y  c3 z  d 3
Then,
a1
b1
c1
d1
b1
c1
D  a2
b2
c2
D1  d 2
b2
c2
a3
b3
c3
d3
b3
c3
Graphics By:- Pradeep
2
Written By:- Raj Kumar Badhan
a1
d1
c1
a1
b1
d1
D2  a2
d2
c2
D3  a2
b2
d2
a3
d3
c3
a3
b3
d3
and ,
x
D
D1
D
, y  2 and z  3
D
D
D
(c)
If D  0, then system has a unique solution and is consistent.
If D = 0 and D1 , D2 and D3 are also zeros, the system is dependant has infinite solutions and
consistent.
If D = 0 and at least one of D1 , D2 and D3 is  0, then system is inconsistent.
11.
A1 
1
 adj  A if A  0
A
i 
 ii 
 AB 
T
 AB 
 B T AT
 iii 
A 
 A 1
(a)
(b)
12.
13.
1
T
1
 B 1 A 1


T
Solution of system of equations using matrices:
Let System be:
a1 x  b1 y  c1 z  d1
a2 x  b2 y  c2 z  d 2
a3 x  b3 y  c3 z  d 3
 a1

A   a2
a
 3
b1
b2
b3
c1 

c2   0
c3 
 x
 
X   y
z
 
 d1 
 
B   d2 
d 
 3
(a)
If |A|  0, then system is consistent and has a unique solution, given by X = A–1B
(b)
If |A| = 0 and (adjoint A). B  0, (adjA). B  0, then system is inconsistent and has no solution.
(c)
If |A| = 0 and (adjoint A). B = 0,then system is either constituent and has infinite solutions Or the
system may have no solution and hence inconsistent.
14.
Solution of system of homogeneous equations:
Let system be:
a1 x  b1 y  c1 z  0
a2 x  b2 y  c2 z  0
a3 x  b3 y  c3 z  0
Graphics By:- Pradeep
3
Written By:- Raj Kumar Badhan
(a)
(b)
a1
b1
c1
It has a trivial solution if a2
a3
b2
c2  0 i.e. x = 0, y = 0 and z = 0 (only solution)
c3
b3
a1
b1
c1
It has non trivial solution if a2
a3
b2
c2  0 , In the case the system has infinitely many
b3
c3
solutions.
Note : Homogeneous system of equations is always consistent i.e. this system has either unique soluti(x=0,
y=0, z=0) or infinitely many solution.
15.
For any square matrix A, A.(adjoint A)=(adjointA).A=|A|.l
GRADED QUESTIONS
LEVEL-1
1.
1 3
Prove that for A = 
 . A + a ’ is a symmetric matrix, where a ’ denotes the transpose of A.
 2 4
3 Marks
2.
Evaluate the following determinants using properties:
1 a
3 Marks
bc
1 b ca
1 c ab
p q
 1
a b
3.
If the points (a, 0), (0, b) and (p, q) are collinear, prove that
4.
Find the area of the triangle with vertices (– 2, 4), (2, –6) and (5, 4).
5.
2
If A = 
4
1
1
Graphics By:- Pradeep
3
 and B =
0
1

0
5

3 Marks
-1 

2  , verify that  AB '  B'A'.
0 
4
Written By:- Raj Kumar Badhan
6.
7.
8.
9.
 2 6 4 


Find A–1 for A=  1 3 2 
 1 5 2


 2 0
0 1
1
1 1
If A = 
 and B = 
 , verify that  AB   B A .
 3 1
 2 4
Solve the following system of equation by matrix method:
4x – 3y = 5
3x – 5y = 1
Using matrix method, solve the system of equations:
x+y+z=1
x – 2y + 3z = 2
x – 3y + 5z = 3
10.
Solve the following system of equations by cramer’s rule:
2y – 3z = 0
x + 3y + 3z = –4
3x + 4y = 3
LEVEL-2
1.
2.
3.
4.
 3 2   4 1
Find a matrix X such that X. 


 1 1  2 3 
 1
w w2 


If A =  w w 2 1  where ‘w’ is one of the imaginary cube root of unity. Find A–1.
 w2 1
w 

Using properties of determinants, evaluate:
xk
x
x
x
xk
x
x
x
xk
Construct a 2 × 2 matrix
A = a ij  where elements are given by
22
(i)
aij   i  2 j  / 2
(ii)
aij 
2
–3i  j
Graphics By:- Pradeep
2
5
Written By:- Raj Kumar Badhan
5.
 cos x  sin x 0 


If f (x) =  sin x cos x 0  then show that f(x) f(y) = f(x + y)
 0
0
1 

6.
 3 1
If f(x) = x² – 5x + 7 and A = 
 , find f  A  .
 1 2 
7.
Find the value of x such that
 1 3 2  1 

1 1  2 5 1 
 2   0
15 3 2  x 

 
8.
9.
1 2 4


Find the symmetric and skew-symmetric parts of the matrix A =  6 8 1 
3 5 7


Solve for the matrix X
 2 1
 5 0   3 9 

X 


 3 2 
 2 4 7 1 
10.
Solve the matrix equation:
 x 2   x   2 
 2   3    
 y  2y  9 
11.
Solve the system of homogeneous equations:
(i)
x+y–z=0
x – 2y + z = 0
3x + 6y – 5z = 0
(ii)
3x – 4y + 5z = 0
x + y – 3z = 0
2x + 3y + z = 0
12.
13.
 1 0 2 


Show that the matrix A =  2 1 2  satisfies the equations A3 – A2 – 3A – l3 = 0 and hence find
 3 4 1


–1
A .
 2 1 3 


If A =  1 1 1  , find A -1 and hence solve the system of equations:
 1 1 1 


2x – y + 3z = 9
x + y +z = 6
x–y+z=2
Graphics By:- Pradeep
6
Written By:- Raj Kumar Badhan
LEVEL-3
1.
2.
3.
4.
1  2n 4n 
 3 4 
n

 , prove that A  
1  2n 
 n
 1 1 
for all n  N. (using principle of MATHEMATICAL INDUCTION)
If A = 
3 Marks
 1 0 1 


Without finding the adj A, find A(adj A) for matrix, A =  3 4
5
 0 6 7 


Hint : A(AdjA) = |A|1
Find K so that the equations
3x – 2y + 2z = 1
2x + y + 3z = –1
x – 3y + kz = 0, may have a unique solution
Show that
1
tan x   1
 tan x   cos 2 x  sin 2 x 
 1

 


1   tan x
1   sin 2 x cos 2 x 
  tan x
5.
 a  ib
If A = 
 c  id
c  id 
 such that a² + b² + c² + d²=1
a  ib 
Then compute A–1.
6.
If a + b + c = 0 and
ax
c
b
c
b x
a
b
a
cx
then show that x = 0 or x =
0
3 2
2
2 
2 a  b  c 




Hint :- a + b + c = 0
a 2  ab  ac  0

1
ab  b 2  bc  0   ab  bc  ca   a 2  b 2  c 2
2
ac  bc  c 2  0 


Graphics By:- Pradeep
7

Written By:- Raj Kumar Badhan
x
7.
If x  y  z and y
z
x 2 1  x3
y 2 1  y 3  0 , then prove that xyz = –1.
z2
1  z3
p b c
8.
If a  p,b  q,c  r , and a
a
Prove that
9.
10.
q c =0
b r
P
q
r


2
pa qb r c
If p + q + r = 0, them prove that
pa
qb
rc
a b c
qc
ra
pb  pqr c a b
rb
pc
qa
b c a
Find the product of the matrices A and B
 5 1 3 
1 1 2




A   7 1 5  , B =  3 2 1 
 1 1 1 
 2 1 3




and use it to solve the equations
x + y + 2z = 1
3x + 2y + z = 7
2x + y + 3z = 2
1
1 
Hint : - AB  4l   A  B  1  B 1  A
4
4 
11.
Use cramer’s rule to solve:
x+y+z=1
ax + by + cz = 7
a²x + b²y + c²z = k²
12.
Using Cramer’s rule, Show that the following system of equations is consistent :
x–y+z=3
2x + y – z = k
-x - 2y + 2z = 1 Also find the solution.
13.
Solve the equation by Cramer’s Rule
(b + c) (y + z) – ax = b – c
(c + a) (z + x) – by = c – a
when a + b + c  0
(a + b) (x + y) – cz = a – b
Graphics By:- Pradeep
8
Written By:- Raj Kumar Badhan
1.
Some more important Questions
3
2
1 1
1
2
If A  
, B = 
,
0
2
0
1
2
1
find the matrix C such that A + B + C is a zero matrix.
2.
Find a 2 × 2 matrix B such that
6 5
11 0 

 B

5 6
 0 11
3.
x yz
2x
2 x`
2y
yzx
2y
2z
2z
zx y
Prove that  
a2
4.
  x  y  z
3
ac  c 2
bc
b2
Prove that a 2  ab
ab
b 2  bc
ac
 4a 2 b 2 c 2
c2
5.
 3 5 
2
If A = 
 show that A  5 A  14l = 0
 4 2 
6.
Using properties of determinants, solve for x:
ax ax ax
ax ax ax 0
ax ax ax
Using properties of determinants, prove that
7.
bc
ca
ab
qr
r p
pq  2 p q
r
yz
zx
x y
z
1 a
1
1
1
1 b
1
1
1
1 c

 0
Let A = 
 tan 


2
Graphics By:- Pradeep
a
x
b c
y
 abc  ab  bc  ca

2
 and 1 be the identity matrix of order 2.
0 

 tan
9
Written By:- Raj Kumar Badhan
 cos   sin  
Prove that 1 + A = (1–A) 

 sin  cos  
8.
 2 3
1 0
If A = 
 and 1= 
.
1 2
0 1
Find  ,  so that A 2  A  1
9.
10.
5 3
A
 satisfies the equation
Show that
 1 2 
x 2  3 x  7  0 and hence find A 1 .
1 2 2


Let A =  2 1 2 
2 2 1


1
find A .Hence prove that A2  4 A  51  0
Solve the following system of equations using Cramer’s rule:
x + y + 3z = 6
x – 3y – 3z = –4
5x – 3y + 3z = 8
11.
12.
Solve the equations by matrix method :
2 3 10
 
4
x y z
4 6 5
  1
x y z
6 9 20
 
2
x y z
13.
14.
1
2
 4


If A =  5 3 3  , find A–1 and hence solve the following equation:
 11 1 7 


4x – 5y – 11z = 12
x – 3y + z = 1
2x + 3y – 7z = 2
2 1 1 


If A =  2 0 1  , find A–1, using A–1solve the following system of linear equation:
 0 2 1 


2x + y + z = 3
2x + z = 5
–2y – z = 1
Graphics By:- Pradeep
10
Written By:- Raj Kumar Badhan
15.
Using matrix method solve the following system of linear equations :
x+y+z=3
2x – y + z = 2
x – 2y + 3z = 2
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
x+y+z=1
x – 2y + 3z = 2
x – 3y + 5z = 3
QUIZ
(Matrices & Determinants)
1.
2.
3.
4.
5.
6.
7.
8.
9.
x  1
 3
If A = 
 is a symmetric matrix, then find the value of x.
2 x  3 x  2
If A is a matrix of order 3 × 4, then each row of A has how many elements ?
Find whether the following matrix is invertible or not.
 2 3
A= 

4 6
(Hint : |A| = 0)
If |A| = |A’| where A’ is transpose of square matrix A, then comment on the type of the matrix ?
Each element of a 3 × 3 matrix A is multiplied by 3 to get matrix i.e. B = 3A if, |B| =  |A| find the
value of  .
Find whether the following system of equations is consistent or not, if consistent find the type of
solutions.
x + 2y = 5
4x + 8y = 20
Find the value of following determinant
52 53 54
53 54 55
5 4 56 57
Find all 2 × 2 real matrices which commute with
1 1
A =

 0 1
a b 
Let B = 
 such that AB = BA (commutative property)
c d 
1 0 
 0 1
 cos sin 
j= 
B= 
If, 1 = 


 prove that,
0 1 
-1 0
-sin cos
B = 1 cos  + j sin 
Graphics By:- Pradeep
11
Written By:- Raj Kumar Badhan
10.
a1
b1
c1
Suppose, D = a2
b2
c2 and
a3
b3
c3
a1  pb1
b1  qc1
c1  ar1
D’ = a2  pb2
b2  qc2
c2  ar2
a3  pb3 b3  qc3
Show that, D’ = D.
c3  ar3
ASSINGMENT
(Matrices & Determinants)
For what values of x and y are the following matrices equal.
3y 
 x  3 y 2  2
2 x  1
A= 
B

y 2  5 y 
6 
 0
 0
(Hint: common solution of y should be considered for answer)
Let,
2
2
1 -2 3 

A= 
B=  –1 2 

 3 2 -1
 4 –5 
Find AB and BA and show that matrix multiplication is not commutative.
2 0 1
Let f(x) = x² – 5x + 6. Find f(A) if A =  2 1 3
1 -1 0
 2 
If, A =  4  B  1 3 – 6
 5 
Verify that, (AB)T = BT.AT
Without expanding, show that
bc
bc
2
c a
2
ca
ca 0
2
2
ab a  b
b2 c 2
a b
(Hint: Apply, R1  R1  a  , R2  R3   c  . Take abc common from C1 & C2)
If points (2, –3), (  , –1) and (0, 4) are collinear find the value of  .
The sum of three numbers is 6. If we multiply is 6. If we multiply the third number by 2 and add it to
first number we get 7. By adding second and third numbers to three times the first number we
get 12. Use determinants to find the number.
(HINT :
x+y+z=6
x + 2z = 7
Graphics By:- Pradeep
12
Written By:- Raj Kumar Badhan
3x + y + z = 12
 3 2
If A  

7 5 
6 7 
B

8 9 
verify that, (AB)–1 = B–1A–1
 2 3
Show that A = 
satisfies the equation x 2  6 x  17  0. Hence find A –1 .

3 4 
1 2 3
Find A–1, where A =  2 3 2  Hence solve the following system of equationsl.
 3 3 4
x + 2z – 3z = –4,
2x + 3y + 2z = 2
and
3x – 3y – 4z = 11
Show that following homogeneous system of equations has non-trivial solutions. Also find the
solutions.
x – 2y + z = 0,
x + y – z = 0,
3x + 6y – 5z = 0
EVALUATION
(Matrices & Determinants)
1.
Find the value of p and q such that
 3 1
A² + pl = qA. Where A = 

 7 5
2.
3.
cos x  sin x 0
1
If f(x) =  sin x cos x 0 . show that f  x    f   x 
 0
0
1 
a 1 0
0
a 0 0 




1
1
0
If A =  0 b 0  , show that A   0 b
 0
 0 0 c 
0 c 1 

For what value of  does the homogeneous system of equation 3x + 2y + z = 0, x + 2y = z,  x +
3y + 3z = 0 has a non-trivial solution.
3 2
–1
5.
If A = 
 , verify that A²– 4A –1 = 0, hence find A .
2
1


6.
Using cramer’s rule, find the function f(x) ax² + bx + c, such that f(0) = 1, f(1) = 2 and f(2) = 7.
7.
Find the values of P, such that the area of the triangle with vertices (5, 4), (–2, 6) and (p, 4) is 35
square units.
Graphics By:- Pradeep
13
Written By:- Raj Kumar Badhan
4.
8.
1 -1 0 
 2 2 –4 


Given A =  2 3 4 and B   4 2 4  , verify that BA = 61, hence solve the system of
0 1 2
 2 1 5 
equation.
X – y = 3,
2x + 3y + 4z – 17 = 0,
y + 2z – 7 = 0
Graphics By:- Pradeep
14
Written By:- Raj Kumar Badhan