(16A005)
Chapter 16
Determinants and Inverses of Square Matrices
Teaching Example 16.5
2 1 3
Evaluate
Teaching Examples
4 2 1
3 0 2
by expanding the determinant
(a) along the 3rd row,
(b) along the 2nd column.
(16A001)
Teaching Example 16.1
3 2
2
 and B   2
Let A  
5 8
x
(a) Evaluate det A.

.
2
2 x  1
1
(b) If det A = det B, find the possible values of x.
(16A006)
Teaching Example 16.6
a m b


Let A   0 1 0  .
c n d


(16A002)
- 1 -
Teaching Example 16.2
a
2b
a 2b
Show that

.
3c  2a 5d  4b
3c 5d
a b
c
d
(16A007)
Teaching Example 16.7
(16A003)
Teaching Example 16.3
3 2 0
Evaluate
By expanding the determinant along a row, show that det A 
2 1 3
1 0 5
9 2 0
and
6 1 3 .
3 0 5
Do you observe any relationship between these two determinants?
0 
 0 2  x
 x 1




 x2
Let A    1 1 2  , B   1 0  2  and C  
3
 2 1 5x 
  1 3  4x 




(a)
(b)
x
.
1 
Expand det A, det B and det C.
If (det C )(det A  det B )  0 , find the values of x.
(16A008)
(16A004)
Teaching Example 16.4
2
5 8
Let A  1
4 7 .
1 3 6
Find the minor and the cofactor of the element 7.
Teaching Example 16.8
a b

Let A   d e
g h

c

b c
a c
a b
f  , verify that g
h
i
 0.
h
i
g
i
g
h
i 
.
(16A009)
(16A013)
Teaching Example 16.9
Teaching Example 16.13
If
Evaluate the following determinants.
a1 a2
a3
b1
c1
b3  8 , find the values of the following determinants.
c3
b2
c2
4
(a)
(a)
a2
b2
c2
a3
a1
b3
b1
c3
c1
b1
(b)
b2
2a1 2a2
 c1  c2
b3
6
1
3
3
1 2 9
 2 4 15
(b)
2

2a3
 c3
2
3
2
1
5
7
3

1
5
2
1
7
(16A014)
Teaching Example 16.14
(16A010)
Teaching Example 16.10
Factorize
1
Let A be a square matrix and det A   .
5
a3
a2
3
2
b
c3
b
c2
a
b .
c
(a) If A is a 3  3 matrix, find det(5A).
(b) If A is a 2  2 matrix, find det(5AT).
(16A015)
- 2 -
Teaching Example 16.15
(16A011)
Teaching Example 16.11
Show that
Expand the following determinants.
3
(a)
2
a 1 a a 1
1
3
3
3
a2 b2 c2
(b)
b 1 b b 1
c 1 c c 1
(16A012)
Evaluate the following determinants without direct expansion.
1
(a)
1
1
2 5 2
4 5 2
(b)
b3
c3
a2
1
b2
1
c 2  (a  b)(b  c)(c  a )(ab  bc  ca) .
1
(16A016)
Teaching Example 16.16
1 0 
 , show that AAT A   I .
(a) Let A  
 0  1
(b)
Teaching Example 16.12
a3
Let B and I be a 2  2 matrix and an identity matrix respectively. If BBT B   I , find the value of
det B.
4 2 5
(16A017)
3 1 6
1 4 2
Teaching Example 16.17
 7  6
1 0
 and I  
 .
Let A  
 4 2 
0 1
(a) Evaluate det( A  I ) .
(b) Find det( A2  I ) and det( A3  I ) .
(c) Hence, or otherwise, find det( A7  I ) .
(16A018)
(16A023)
Teaching Example 16.18
Teaching Example 16.23
Let A be a square matrix such that A  5 A  2 I  0 . Show that A is non-singular.
Let A and B be two non-singular matrices of the same order.
(16A019)
(a) Show that A1B 2 A is non-singular.
(b) Show that if A2  5 A  I , 5 I  A is also non-singular.
2
Teaching Example 16.19
Find the cofactor matrix of each of the following.
 2 3

(a) 
 0 5
 3 1 2


(b)  1  1 0 
0 2 1


(16A024)
Teaching Example 16.24
2 1
 .
Let A  
 3 2
(a) Show that A is a non-singular matrix.
(b) Show that A2  4 A  I  0 and hence find A1.
(c) Is it true that ( A 1 ) 2  4 A1  I  0 ?
(16A020)
Teaching Example 16.20
2 5 
 . Show that A is non-singular and find A1.
Let A  
 4 11
- 3 -
(16A021)
Teaching Example 16.21
(16A025)
Teaching Example 16.25
  2 0
 2 0
 and B  
 .
Let A  
 1 1
 2 6
(a) Show that A is a non-singular matrix and find the inverse of A.
(b) Show that A1BA is a diagonal matrix.
1 2 1 


Let A   0 1 3  .
 2 1  1


n
(c)
(a) Show that A is non-singular and find A1.
 2 4 1


(b) If B   1 0 2  , find a matrix C such that CA  B.
 1 1 2


 xn
x 0
  
It is given that 
0
y


0
0
 for any real numbers x and y and positive integer n. Use the
y n 
result in (b) to evaluate B10.
(16A026)
Teaching Example 16.21 (Extra)
Teaching Example 16.22
 1

Let A    3

 0
Determine whether each of the following matrices is singular or not, where x and y are non-zero real
(a) Show that A is non-singular and find the inverse of A.
numbers.
 1  3 0


(b) If B    3
1
0  , find a matrix C such that AC  B  I .


0
0
4


(16A022)
 2 2x 

(a) 
 5  5x 
1 x 1 y 


(b)  2 2 x  1 y 
 3 3x  1 y 


3 0

1 0 .

0 2 
Basic Questions
(16A027)
Teaching Example 16.24 (Extra)
 3  2
 .
Let A  
 4 3 
§16.1 Determinants
(16B001)
(a) Show that A is a non-singular matrix.
(b) If A2  xA  yI  0 , find the values of the real numbers x and y, and hence find A1.
(c) For the equation A  xA  yI  0 obtained in (b), is it true that ( A )  xA  yI  0 ?
Evaluate the following determinants by definition.
6 2
(a)
(b)
0 1
(16A028)
(16B002)
Teaching Example 16.25 (Extra)
 2  1
1  17  2 
 and B  
.
Let A  
7   6 18 
3 2 
(a) Show that A is a non-singular matrix and find the inverse of A.
Evaluate the following determinants by definition.
1 2
2
1
2
15
(a)
3
10
(b)
(b) Show that A1BA is a diagonal matrix.
n
(c)
 an
 a 0
  
It is given that 
0
b


0
0
 for any real numbers a and b, and positive integer n. Use the
b n 
2 7
3 6
- 4 -
2
3
1

4
3
5
5

6
2
0
(16B003)
Evaluate the following determinants by Sarrus’ rule.
result in (b) to evaluate B5.
2
(a)
1
3
1 3 4
0 1  2
(b)
1
4 1 2
3  2 3
(16B004)
Evaluate the following determinants by Sarrus’ rule.
1 1
1
2 4
2
1 1
(a)
1
(b)
3
2 4
4
1
1
1
4
2
2
2
3
3
4
4
(16B005)
Let det A 
(a)
11 24
30
5
. Find the minors and the cofactors of
the element 11,
(b)
the element 30.
(16B006)
(16B012)
1
2
Let det A  0  3 1 . Find the minors and the cofactors of
2
(a)
3
y
 2y
y2
0
1
y
3
Expand and factorize
5
the element 0,
(b)
Using cofactor expansion, evaluate
(a)
along the first row,
(b)
along the third column.
1
(16B013)
x x3
Solve
 0.
1
2
2
4 3 0
5 3 4
by expanding
(16B008)
- 5 -
Using cofactor expansion, evaluate
 3
5
(a)
along the first column,
(b)
along the second row.
(16B009)
1 x  2
Expand
.
x
2
(16B014)
Solve
2
2
1
4
0
2
3
by expanding
1 2 x2
 1 .
2x 3
(16B015)
d 
a b 
 c
 and B  
 . Show that det A  det B .
Let A  
c d
 a  b
(16B016)
 x 2 0


 x  y 0
 . Show that det A  x  det B  0 .
Let A   0 y x  and B  
 x  3y 2
 x 0 2


§16.2 Properties of Determinants
(16B017)
(16B010)
0
5
Expand
2x
3x 4
1
Evaluate
x
.
x 2
(16B011)
Expand
2
2 1
3 4
2
1 .
5
(16B018)
x
2
x
1 1 x 0 .
1
x
3
1 .
1
the element 5.
(16B007)
0
5
4 1 5
Evaluate
4 5 4 .
2 4 1
(16B019)
(16B024)
5
Evaluate
3
3 3
8 6
2
1 p
2 .
4
3
4
4  12 8 .
1
0 2
Expand and factorize
(16B021)
4
6
4 7  10 .
5 1 7
Expand and factorize
- 6 -
a3
b1
c1
b3  2 , find the values of the following determinants.
c3
(a)
1
a2
a3
a1
b2
c2
b3
c3
b1
c1
c2
(b)
b2
Show that
a2
(16B023)
(a)
x
y
x .
y
1 a 1 .
1 1 a
a1
a2
a3
b1
c1
b2
c2
b3  1 , find the values of the following determinants.
c3
2a2
3a1 4a3
2b2
2c2
3b1
3c1
4b3
4c3
a2
7a3
c2
7c3
4 3
7  18
 10
 2  0 without direct expansion.
12
sec 2 
tan 2  1
sec2 
sec 2 
tan 2  1  0 without direct expansion.
tan 2  1
(16B029)
Let A and B be two 2  2 matrices and I be an identity matrix. If AB  2 I , show that det A 
a1  2a3 c1  2c3 b1  2b3
(b)
15
(16B028)
 c1  b1  a1
c3
b3
a3
Show that
If
y
x
(16B027)
a1 a2
b2
c2
x
a 1 1
(16B022)
If
0
(16B026)
2
Evaluate
x
(16B025)
3
Evaluate
1 r
r p .
pq
Expand and factorize 1 q
(16B020)
qr
b2
7b3
(16B030)
Let A be a 3 3 matrix. If AT  A  0 , show that det A  0 .
4
.
det B
§16.3 Inverses of Square Matrices
(16B036)
(16B031)
Find the cofactor matrix of each of the following matrices.
Determine whether each of the following matrices is singular or non-singular.
6 
 7 9
 4


(a) 
(b) 
  2 5
  6  9
(a)
  3  2


 2
3 

(b)
(16B032)
Determine whether each of the following matrices is singular or non-singular.
 ab  b 
 , where a and b are real constants
(a) 
 a 1 
b 
 a
 , where a and b are positive constants and a  b
(b) 
b  a a  b

1

1
4
1

3
1
2
1
5
1
2
1

3
1
4

1

(16B037)
Find the adjoint matrix of each of the following matrices.
(a)
2 4 


 0  1
(b)
(16B033)
1 4 1


0 0 7
 4 6 9


Determine whether each of the following matrices is singular or non-singular.
(a)
- 7 -
 0 1 3


 9 2 6
  4 0 0


(b)
3 9
 2


 4 5 2 
  2 19 23 


(16B038)
Find the adjoint matrix of each of the following matrices.
(a)
  3 0


 1 6
(b)
(16B034)
4 
1 3


2  5 0 
 1 0  2


Determine whether each of the following matrices is singular or non-singular.
(a)
 2 0  12 


 5 9  30 
 1 3
6 

(b)
8 
1 1


 3  2  2
 5 10 5 


(16B039)
Show that the following matrices are non-singular and find their inverses.
(a)
 5  7


  3  4
(b)
(16B035)
2 2 1 


1 0 1 
 0 1 1


Find the cofactor matrix of each of the following matrices.
(a)
5 4 


 2 1
(b)
 5 1 3


 2  1 1
 0  2 1


(16B040)
Show that the following matrices are non-singular and find their inverses.
(a)
 9 16 


 25 36 
(b)
  3 4  6


 5  10 0 
 1 3  2


Section A Questions
(16B041)
2 
x
 . Find the range of values of x such that A is non-singular.
Let A   2
 x 1 x 
§16.1 Determinants
(16B042)
Show that


 1 m 8 
Let M    2 0 m  . Show that M is non-singular for all real values of m except 2.

3
m 1

2

(16C001)
x
x2 1 x

2
y  x2
y 1 y
y
.
x  y2
ab  5cd
d  10b
cd
2b
(16C002)
Show that
ab
d
cd
2b

.
(16B043)
Let B be a square matrix such that B 2  7 B  9 I  0 . Show that B is non-singular and find B 1 in terms
of B and I.
(16C003)
Show that
(16B044)
Let Y be a square matrix such that Y 3  Y  2 I  6Y 2 . Show that Y is non-singular and find Y 1 in terms
of Y and I.
- 8 -
(16B045)
 n 1 
 , where  and  are the roots of the equation x2 + x + k = 0 and n  0 . Find the values
Let An  
n
1



of k so that A0 + A1 + A2 is singular.
x 1
x 2
x 2
 0 for all real values of x.
2x
sin 
(sin   cos ) 2 sin 2  2

.
1

1
cos
2
(16C004)
Show that
1
sin 2
(16C005)
Show that
 tan x
cos x
sin x cos x  cos x cos x  sin x
2
(16B046)
It is given that A and B are two non-singular square matrices of the same order. Show that
det( ABA1B 1 )  1 .
(16C006)
Solve
(16B047)
Let X be a non-singular matrix. If X 1  X T , show that ( X T X ) 1  I .
1
x  3.
1 x
2
(16C007)
Solve
7
1  x2 1  x
1 x x
.

6
2 2
1

x
2
x
 1.
(16C008)
(16C012)
x2
(a)
2
2
Expand
2x
x 2
2 x
x2
(b)
Hence, solve
2x
2
2
2x
by Sarrus’ rule.
2x
 x 2  8.
2 x
(16C009)
(a)
(b)
Expand
x
x 1
1
x 1
1
2
1
3
4
Hence, solve
- 9 -
x
x 1
x 1
1
2
1
by cofactor expansion.
1  2 k 
5 4 1 




Let M   k 3 0  and N    1 0 k  .
 0 1 2 
 k 2  3




(a)
Find M + N.
(b)
If det(M + N) = –14, find the value(s) of k.
(16C013)
0 
 2 x
2
 and B  
 .
Let A  
 6 1
 3 x  1
(a) Find AB.
(b)
If det(AB) = –16, find the value(s) of x.
(16C014)
1
3  0.
4
(a)
(b)
Solve the quadratic equation 2x2 + 3x – 2 = 0.
Hence, find the value(s) of a such that 2
(16C010)
(16C015)
 a 1 5 


Let A   2a 0  2  .
 3a 2
1 

 log x log 2 log 5 


Let P   2
2
1 .
 4
0
 2 

(a)
Find det A.
(a)
Find det P.
(b)
If det A = 16, find the value(s) of a.
(b)
If det P = 0, find the value(s) of x.
a
2
a 1 3
2
3
(16C011)
(16C016)
x
x 1
1


Let B  1 x  1 x  2 .
1 2 x
x 2 

 x

Let P   x 2
3

(a)
Using cofactor expansion, factorize det B.
(a)
Expand det P by cofactor expansion.
(b)
Hence, solve det B  0.
(b)
Find the value of x such that det P attains its minimum.
0  2 x  1

1
x3  .
0
x  1 
a
2
a 1 3
2  0.
(16C017)
(16C022)
If ,  and  are the interior angles of a triangle, show that
1
0
sin 
cos 
 sin 
cos 
1
a  b  2c
Show that
sin   sin   sin  .
cos 
(16C018)
a b
 , where a and b are real numbers. If det Y = 0, prove that Y n1  (2a )n Y for any positive
Let Y  
b
a


integer n.
c
c
a
b
2a  b  c
b
 2(a  b  c)3 .
a
a  2b  c
(16C023)
Expand and factorize
a2
(b  c) 2
bc
2
(c  a )
( a  b) 2
ca .
ab
b
c2
2
(16C019)
(a)
For any 2  2 matrix [aij]22, if a11 = a22 and a12 = a21, then the matrix [aij]22 is called a
(16C024)
2  2 bisymmetric matrix. Consider two 2  2 bisymmetric matrices P and Q.
(i)
(ii)
- 10 -
(b)
Show that PQ is also a 2  2 bisymmetric matrix.
Verify that det( PQ )  det P  det Q .
 1  2  3  4  2


Let R  

2
1

4
3


  2
 2  3  3 

.
2   3
3 
Using the result of (a), find det R.
1 a a 2  bc
Simplify 1 b b 2  ca , where a, b and c are non-zero constants.
1 c c 2  ab
(16C025)
 2 3
If  and  are the roots of x2  3x  2  0, show that  2 3
§16.2 Properties of Determinants
1
(16C020)
2
2
 2  0 without direct expansion.
3
ln 2 ln 10 1
Show that
ln 3 ln 15 1  0 without direct expansion.
ln 5 ln 25 1
(16C026)
If k1, k2, k3, k4, k5, k6, k7, k8, k9 is a geometric sequence, show that
(16C021)
Expand and factorize
y
x
x y
x
x y
x y
y
y
x
.
k1
k4
k7
k2
k3
k5
k6
k8  0 .
k9
(16C027)
5 6 
 1 0
 and I  
 .
Let A  
 4 10 
0 1
(a) Evaluate det(A  2I).
(b) Find the values of
(i)
det(A2  4I),
(ii)
det(8I  A3).
(b)
30 10 10
(i)
 22 32 
 1 0
 and I  
 . It is given that det(M  kI) = 0, where k  0 .
Let M  
 75 98 
0 1
(a) Find the value(s) of k.
Find the values of
(i)
(c)
2012 1988
20 30 20
10 10 30
(ii)
(16C031)
(16C028)
(b)
Hence, evaluate the following determinants.
(a)
x
y
z
ya
z
1002 2002 3000
(ii) det(k3I  M3).
Deduce a value of m such that det(M999  mI) = 0, where m  0 .
- 11 -
(16C029)
(16C032)
3 0 1 
1 0 0




Let B   1 2  1 and I   0 1 0  .
3 1 1 
0 0 1




 2 3 5 


Let B   0 1  3  .
 1 2 0 


Evaluate det(B I).
Show that
(i)
det(B2 I) = 0,
(ii)
det(I  B3) = 0,
(iii) det(B
2012
(16C030)
ab
(a)
Expand and simplify
(a)
Show that B is non-singular.
(b)
Find B1.
(16C033)
I) = 0.
b
ab
a
ab
a .
a b
b
ab
y .
za
Hence, evaluate 1002 2000 3002 .
§16.3 Inverses of Square Matrices
(a)
(b)
x
1000 2002 3002
(b)
det(M2 k2I),
Expand and simplify
xa
 2 0 3


Let P   1  1 1  .
 3  2 1


(a)
Show that P is non-singular.
(b)
Find P1.
12
1988 2012 12
2000 2000 2012
(16C034)
(16C039)
5  4
 3  2 1
 12




Let A   6 4 0  and B    18 5
6 .
 4 1 3
  22  5 24 




 1 2  2


Let M   0 1 0  .
3 3 1 


(a)
Find AB and BA.
(a)
Show that M 3  3M 2  9 M  7 I  0 .
(b)
Hence, find A1.
(b)
Hence, or otherwise, find M .
(16C035)
8
 7
 5 2 3
 2




Let X    1 3 1  and Y    1 19  8  .
  3 4 2
 5  26 17 




(a)
Find XY and YX.
(b)
Hence, find X1.
- 12 -
(b)
Determine whether each of the matrices obtained in (a) is non-singular. If yes, find its inverse.
(16C037)
8 6
 .
Let D  
9 4
(a) Show that D 2  12 D  22 I  0
(b)
(c)
(16C040)
4 0 
 .
Let A  
 5  1
(a) Find A1.
(b)
(16C036)
5
3 
0 1 1
2
 and Q  
 .
Let P  
 2 0 2
 1  3  2
(a) Find PQT and QTP.
1
Find D .
Determine whether ( D 1 ) 2  12 D 1  22 I  0 is true.
(16C038)
2 0
0


Let P    1 1 1  .
 0  3 2


(a)
Evaluate P 3  3P 2  7 P .
(b)
Hence, or otherwise, find P .
1
1
 4 0  x   6 
     .
Hence, find the values of x and y such that 
 5  1 y    7 
(16C041)
 12  5 
 .
Let B  
  21 8 
(a) Find B1.
(b)
a b
 such that
Hence, find a matrix 
c d 
 12  5  a b   9 4 


  
 .
  21 8  c d   0 5 
(16C042)
 3 4 2 


Let M   5
0
1 .
 1 5  2


(a)
Find M 1.
(b)
 3  4 2  x   6 

   
Hence, find the values of x, y and z such that  5
0
1  y    8  .
  1 5  2  z   3 

   
(16C043)
2 0 
 and A2 + A = I, where  and  are constants.
Let A  
0

3


(a) Find the values of  and .
(b) Using the result of (a), find A–1.
1 2
 .
(c) Find B such that ABA  
3 4
(16C044)
1 3 
 2  9
 , B  
 and Y  A1BA .
Let A  
 0  1
0 5 
(a) Find A1.
(b) Show that Y is a diagonal matrix, and hence, find Y 5.
(c)
Using the results of (a) and (b), find B5.
- 13 -
(16C045)
 2  4
6  2
 2 0
 , Q  
 and M  
 .
Let P  
 4  6
6 1
 0 3
(a) Find P1.
(16C048)
Let A be a 2 2 non-singular matrix. Show that det (aI  A1 ) 
(16C049)
 cos 
Let A  
 sin 
 sin  
.
cos  
(a)
Find A 1 .
(b)
 cos( n )  sin( n ) 
 for any positive integer n.
Prove by mathematical induction that ( A 1 ) n  
 sin( n ) cos( n ) 
(16C050)
Let A be a 2  2 non-singular matrix. It is given that det A  0 .
m  n
 , where m and n are real numbers and m 2  n 2  1 .
Show that A1  AT if and only if A  
n m 
(16C051)
A and B are two 2 2 non-singular matrices. It is given that A–1 = AT, B–1 = BT and det A + det B = 0.
(a)
Show that (det A) 2 = 1. Hence, show that det(AB) = –1.
(b)
Show that M  P1QP.
(b)
Show that A + B = A(B + A)TB.
(c)
(d)
Find M100, and hence, find P1Q100P.
Using the above results, find Q100.
(c)
Hence, show that A + B is singular.
(16C046)
 0 a b 


Let M   a
0  c .
 b c
0 

(a)
Show that M 3  (a 2  b 2  c 2 ) M .
(b)
If M is non-singular, evaluate det[ M 2  (a 2  b 2  c 2 ) I ] .
(16C047)
Let A and B be two square matrices of the same order, where A is non-singular. Show that
det ( I  A 1 BA)  det ( I  B ) .
a2
1 

 det  A  I  .
det A
a 

Section B Questions
(16D004)
(a)
(16D001)
1 2
 .
Let A  
3 4
(a) Find (A + I)–1 and (A – I) –1.
(b)
(c)
Let A be a 2 2 matrix such that A2  A  I  0, where I is the 2 2 identity matrix.
(i)
Find A1.
(ii)
Show that A3  I.
(iii) Show that
100
A
k
  A 1 .
k 0
(i)
(A2 – I)–1,
(ii)
(A–1 + I)–1.
 1 1 
 .
Let B  
 1 0
(i) Find B, and hence, find B1.
(i)
Expand (A – I)(A–1 + I).
(ii)
(ii)
Hence, find det(A – A–1).
Using the result of (a), find
(b)
Find B, B100 and B101.
(16D005)
(16D002)
 a 1 a 
 , where a is a real number.
Let A  
1  a  a 
(a) Find the range of values of a such that A is non-singular.
- 14 -
(b)
Find A2. Hence, find A–1.
(c)
Simplify
(i)
(ii)
(I + A)6,
( A 1  2 A 3 ) 1 ,
(iii)
A  A2  A3  ...  A2012 .
Let A and B be two 2  2 non-singular matrices such that A2 = B, where det A > 0.
(a) Show that
(i)
AB = BA,
AB 1  A 1
( BA1  AB 1 ) 2  B  B 1  2 I
 4 0
 , find
If B  
 3 1
(i) A,
(ii)
(iii)
(b)
(ii)
–1
B ,
–1
–1 2
(iii) (BA + AB ) .
(16D003)
Let A be a square matrix such that A2 – ( + )A + I = 0.
(a)
(b)
(i)
(ii)
Show that either det(A – I) = 0 or det(A – I) = 0.
A student claims the following.
“The result in (a)(i) implies that either A – I or A – I is a zero matrix.”
Determine whether the student’s claim is true.
1 4
 .
Let B  
 2 3
(i)
Using the result of (a)(i), find the values of  and  such that B 2  (   ) B   I  0 ,
where    .
(ii)
Hence, find B and B4 without direct expansion.
–1
(16D006)
 1 0 0 


Let A   3 1 1  .
  6  2  1


(a)
Find A2 and A4.
(b)
Find det(A2000).
(c)
Suppose P and Q are two 3  3 non-singular matrices such that P–1QP = A.
(i) Find Q2000.
(ii)
Find det(Q2001).
(16D007)
 2 1
 1  1
 , B  
 and C = B–1AB.
Let A  

1
0

1
2




(a) Find C.
(b)
(c)
(d)
n
For a positive integer n, guess a formula for C . Prove your assertion by mathematical induction.
Hence, or otherwise, find
n
(i)
A,
(ii)
(A – C ) .
n
n –1
Find
n
(i)
P,
(ii)
det(P – R ).
n
n
(16D010)
 0 1 1

1 
Let M 
 1 0 1 .
3

  1  1 0
(16D008)
(a)
Show that M3 + M = 0.
 3 1
 cos   sin  

 and P  
 , where 0    .
Let M  
2
 1 3
 sin  cos  
1

(a) Find P .
 3  sin 2 cos 2 
 .
(b) Show that P 1MP  
 cos 2 3  sin 2 
 a 0
 , find
(c) If P 1 MP  
 0 b
(b)
Find det M and det (M3).
(c)
(d)
Show that (I + M + M2)–1 = I – M + M2.
Let X be a 3  3 matrix such that ( I  M ) X  M 2 ( I  X ) . Using the above results, find X.
- 15 -
(i)
(ii)
the values of , a and b,
Mn.
(16D011)
1 n a
1  n b 




Let A   0 1 n  and B   0 1  n  .
0 0 1
0 0
1 



(a)
Show that AB = BA.
(b)
1 1 0


Let M   0 1 1  .
0 0 1


(16D009)
 4 0 1
 1 1 0




Let P   2 3 2  , Q   2 0 1  and R  Q 1PQ .
 1 0 4
 1  1 0




(i)
–1
(a)
Find Q .
(b)
5 0 0


Show that R   0 3 0  .
 0 0 3


(c)
5

Show that R   0
0

n
n
0
n
3
0
0

0  for any positive integer n.
3n 
(c)
Find M2, M3 and M4.
(ii) Guess a formula for Mn for all n  1. Prove your assertion by mathematical induction.
Using the above results, find the inverse of Mn.
(16D012)
 p1 
 
Let A be a non-zero 3  3 matrix and P   p 2  be a non-zero 3  1 matrix.
p 
 3
Assume that AP = 0 and A2 = A.
(a)
M2V3 Chapter 16 Quiz
Show that
(i)
A is singular,
Determinants and Inverses of Square Matrices
(ii) An  A , where n is any positive integer.
Suppose (I + A)–1 = I + A for some real constants  and  , where   –1.
(b)
(i)
Show that   

 1
.
Solve the matrix equation (I + A)X = P, where X is a non-zero 3  1 matrix.
1 

(iii) Solve the matrix equation    A 
I Y  P , where Y is a non-zero 3  1 matrix.
 1 

Name: ______________________
Class: ____________ (
)
Result: ____________
(ii)
1
4
4
2
5
8
2
0 .
4
1.
Evaluate
2.
In each of the following, find the values of x.
1
2
(a)
0
(b)
x  7 1  x2
- 16 -
3.
Expand
(a)
0
2
1
y 1
y
x 1
0
1 .
0
b
b2
sin x
c
c2
cos x
2
Hence, solve cos x  2 sin x  2 sin x  cos x  0 , where 0  x < 2 .
sin 2 x
5.
1
Show that b  c c  a a  b  (a  b)(b  c)(c  a)(a  b  c) .
a2
(b)
x4
2
y
a
4.
x
cos 2 x
2
Find the inverses of the following matrices.
(a)
 4  12 


 2  6 
(b)
1 
1 1


5 
0 2
 7 1  3


6.
7.
- 17 -
8.
It is given that p 1, –2.
(a)
p

Show that  1
1

(b)
p

Find  1
1

1
p
1
1
p
1
1

1  is non-singular.
p 
1
1

1 .
p 
2
1 1
 4 1 5 




Let P   2  1 3  and Q  11  7 1  .
 3  1  1
 1 4  3




(a)
Find PQ and QP.
(b)
Hence, find Q1.

 0

Let A    4
 0


1
2
3
0
1

1 0 1 
2


2  and P  1 1 4  .
1  1 0 
1 



(a)
Find P1.
(b)
(c)
Show that P1AP is a diagonal matrix.
Hence, or otherwise, find An, where n is a positive integer.
~ End of Quiz ~