Matrices and Determinants

Advanced Level Pure Mathematics

Advanced Level Pure Mathematics

Chapter 8Matrices and Determinants

8.1Introduction : Matrix / Matrices2

8.2Some Special Matrix3

8.3Arithmetrics of Matrices4

8.4Inverse of a Square Matrix16

8.5Determinants19

8.6Properties of Determinants21

8.7Inverse of Square Matrix by Determinants27

8.1Introduction : Matrix / Matrices

1.A rectangular array of mn numbers arranged in the form

is called an mn matrix.

e.g. is a 23 matrix.

e.g. is a 31 matrix.

2.If a matrix has m rows and n columns, it is said to be order mn.

e.g. is a matrix of order 34.

e.g. is a matrix of order 3.

3. is called a row matrix or row vector.

4. is called a column matrix or column vector.

e.g. is a column vector of order 31.

e.g.is a row vector of order 13.

5.If all elements are real, the matrix is called a real matrix.

6. is called a square matrix of order n.

And is called the principal diagonal.

e.g. is a square matrix of order 2.

7.Notation :

8.2Some Special Matrix.

Def.8.1If all the elements are zero, the matrix is called a zero matrix or nullmatrix, denoted by .

e.g. is a 22 zero matrix, and denoted by .

Def.8.2Let be a square matrix.

(i)If for all i, j, then A is called a zero matrix.

(ii)If for all i<j, then A is called a lower triangular matrix.

(iii)If for all i>j, then A is called a upper triangular matrix.

i.e.Lower triangular matrixUpper triangular matrix

e.g. is a lower triangular matrix.

e.g. is an upper triangular matrix.

Def.8.3Let be a square matrix. If for all , then A is called a diagonal matrix.

e.g. is a diagonal matrix.

Def.8.4If A is a diagonal matrix and , then A is called an identity matrix or a unit matrix, denoted by .

e.g. ,

8.3Arithmetrics of Matrices.

Def.8.5Two matrices A and B are equal iff they are of the same order and their corresponding elements are equal.

i.e. .

e.g..

N.B. and

Def.8.6Let and .

Define as the matrix of the same order such that for all i=1,2,...,m and j=1,2,...,n.

e.g.

N.B.1. is not defined.

2. is not defined.

Def.8.7Let . Then and A-B=A+(-B)

e.g.1If and . Find -A and A-B.

Thm.8.1Properties of Matrix Addition.

Let A, B, C be matrices of the same order and O be the zero matrix of the same order. Then

(a)A+B=B+A

(b)(A+B)+C=A+(B+C)

(c)A+(-A)=(-A)+A=O

(d)A+O=O+A

Def.8.8Scalar Multiplication.

Let , k is scalar. Then kA is the matrix defined by .

i.e.

e.g.If ,

then -2A=;

N.B.(1)-A=(-1)A

(2)A-B=A+(-1)B

Thm.8.2Properties of Scalar Multiplication.

Let A, B be matrices of the same order and h, k be two scalars. Then (a) k(A+B)=kA+kB

(b)(k+h)A=kA+hA

(c)(hk)A=h(kA)=k(hA)

Def.8.9Let . The transpose of A, denoted by , or , is defined by

e.g. , then

e.g. , then

e.g., then

N.B.(1)

(2) , then

Thm.8.3Properties of Transpose.

Let A, B be two mn matrices and k be a scalar, then

(a)

(b)

(c)

Def.8.11A square matrix A is called a symmetric matrix iff .

i.e.A is symmetric matrix

e.g. is a symmetric matrix.

e.g. is not a symmetric matrix.

Def.8.12A square matrix A is called a skew-symmetric matrix iff .

i.e.A is skew-symmetric matrix

e.g.2Prove that is a skew-symmetric matrix.

e.g.3Is for all i=1,2,...,n for a skew-symmetric matrix?

Def.12Matrix Multiplication.

Let and . Then the product AB is defined as the mp matrix where

.

i.e.

e.g.4Let .Find AB and BA.

e.g.5Let . Find AB. Is BA well defined?

N.B.In general, AB  BA .

i.e.matrix multiplication is not commutative.

Thm.8.4Properties of Matrix Multiplication.

(a)(AB)C = A(BC)

(b)A(B+C) = AB+AC

(c)(A+B)C = AC+BC

(d)AO = OA = O

(e)IA = AI = A

(f)k(AB) = (kA)B = A(kB)

(g).

N.B.(1)Since AB  BA ;

Hence, A(B+C)  (B+C)A and A(kB)  (kB)A.

(2).

(3)

e.g.Let

Then

But A  O and B  C,

so .

Def.Powers of matrices

For any square matrix A and any positive integer n, the symbol

denotes .

N.B.(1)

(2)If , then

e.g.6Let , ,and

Evaluate the following :

(a)(b)

(c)(d)

e.g.7(a)Find a 2x2 matrix A such that

.

(b)Find a 2x2 matrix such that

and .

(c)If , find the values of .

e.g.8Let . Prove by mathematical induction that

[HKAL92](3 marks)

e.g.9(a)Let where .

Prove that for all positive integers n.

(b)Hence, or otherwise, evaluate .[HKAL95] (6 marks)

e.g.10(a)Let and B be a square matrix of order 3. Show that if A

and B are commutative, then B is a triangular matrix.

(b)Let A be a square matrix of order 3. If for any , there exists such that , show that A is a diagonal matrix.

(c)If A is a symmetric matrix of order 3 and A is nilpotent of order 2 (i.e. ), then A=O, where O is the zero matrix of order 3.

Properties of power of matrices :

(1)Let A be a square matrix, then .

(2)If , then

(a)

(b).

(3)

e.g.11(a)Let X and Y be two square matrices such that XY = YX.

Prove that (i)

(ii) for n = 3, 4, 5, ... .

(Note:For any square matrix A , define .)(3 marks)

(b)By using (a)(ii) and considering , or otherwise, find

.(4 marks)

(c)If X and Y are square matrices,

(i)prove that implies XY = YX ;

(ii)prove that does NOT implies XY = YX .

(Hint : Consider a particular X and Y, e.g., .)

[HKAL90](8 marks)

8.4Inverse of a Square Matrix

N.B. (1) If a, b, c are real numbers such that ab=c and b is non-zero, then

and is usually called the multiplicative inverse of b.

(2)If B, C are matrices, then is undefined.

Def.A square matrix A of order n is said to be non-singular or invertible if and only if there exists a square matrix B such that AB = BA = I.

The matrix B is called the multiplicative inverse of A, denoted by

i.e..

e.g.12Let , show that the inverse of A is .

i.e..

e.g.13Is ?

Def.If a square matrix A has an inverse, A is said to be non-singular or invertible. Otherwise, it is called singular or non-invertible.

e.g. and are both non-singular.

i.e.A is non-singular iff exists.

Thm. The inverse of a non-singular matrix is unique.

N.B.(1),so I is always non-singular.

(2)OA = O  I , so O is always singular.

(3)Since AB = I implies BA = I.

Hence proof of either AB = I or BA = I is enough to assert that B is the inverse of A.

e.g.14Let .

(a)Show that .

(b)Show that A is non-singular and find the inverse of A.

(c)Find a matrix X such that .

Properties of Inverses

Thm.Let A, B be two non-singular matrices of the same order and  be a scalar.

(a).

(b) is a non-singular and .

(c) is a non-singular and .

(d)A is a non-singular and .

(e)AB is a non-singular and .

ProofRefer to Textbook P.228.

8.5Determinants

Def.Let be a square matrix of order n. The determinant of A, detA or |A| is defined as follows:

(a)If n=2,

(b)If n=3,

or

e.g.15Evaluate (a)(b)

e.g.16If , find the value(s) of x.

N.B.

or

or ......

By using

e.g.17Evaluate (a)(b)

8.6Properties of Determinants

(1)i.e..

(2)

(3)

(4)

(5)If , then

(6)

(7)

N.B.(1)

(2)If the order of A is n, then

(8)

N.B.

e.g.18Evaluate (a) ,(b)

e.g.19Evaluate

e.g.20Factorize the determinant

e.g.21Factorize each of the following :

(a)[HKAL91] (4 marks)

(b)

Def.Multiplication of Determinants.

Let ,

Then

Properties :

(1)det(AB)=(detA)(detB)i.e.

(2)|A|(|B||C|)=(|A||B|)|C|N.B.A(BC)=(AB)C

(3)|A||B|=|B||A|N.B.ABBA in general

(4)|A|(|B|+|C|)=|A||B|+|A||C|N.B.A(B+C)=AB+AC

e.g.22Prove that

Minors and Cofactors

Def.Let , then , the cofactor of , is defined by , , ... , .

Since+

Thm.(a)

(b)

e.g.,, etc.

e.g.23Let and be the cofactor of , where.

(a)Prove that

(b)Hence, deduce that

8.7Inverse of Square Matrix by Determinants

Def.The cofactor matrix of A is defined as .

Def.The adjoint matrix of A is defined as

.

e.g.24If , find adjA.

e.g.25(a)Let , find adjA.

(b)Let , find adjB.

Thm.For any square matrix A of order n ,

A(adjA) = (adjA)A = (detA)I

Thm.Let A be a square matrix. If detA  0 , then A is non-singular and .

ProofLet the order of A be n , from the above theorem ,

e.g.26Given that , find .

e.g.27Suppose that the matrix is non-singular , find .

e.g.28Given that , find .

Thm.A square matrix A is non-singular iff detA  0 .

e.g.29Show that is non-singular.

e.g.30Let , where .

(a)Find the value(s) of x such that A is non-singular.

(b)If x=3 , find .

N.B.A is singular (non-invertible) iff does not exist.

Thm.A square matrix A is singular iff detA = 0.

Properties of Inverse matrix.

Let A, B be two non-singular matrices of the same order and  be a scalar.

(1)

(2)

(3)

(4) for any positive integer n.

(5)

(6)The inverse of a matrix is unique.

(7)

N.B.

(8)If A is non-singular , then

N.B.

(9)If A is non-singular , then

(10)

(11) If , then .

(12) If , then where n  0 .

e.g.31Let , and .

(a)Find and .

(b)Show that .

(c)Hence, evaluate .

e.g.32Let and .

(a)Find .

(b)Find , where n is a positive integer.[HKAL94] (6 marks)

e.g.33(a)Show that if A is a 3x3 matrix such that , then detA=0.

(b)Given that ,

use (a) , or otherwise , to show .

Hence deduce that .[HKAL93] (7 marks)

e.g.34(a)If  ,  and  are the roots of , find a cubic equation whose

roots are .

(b)Solve the equation .

Hence, or otherwise, solve the equation

.[HKAL94] (6 marks)

e.g.35Let M be the set of all 2x2 matrices. For any ,

define .

(a)Show that for any A, B, C  M and ,  R,

(i),

(ii),

(iii)the equality “” is not necessary true.

(5 marks)

(b)Let A  M.

(i)Show that ,

where I is the 2x2 identity matrix.

(ii)If and , use (a) and (b)(i) to show that

A is singular and .(5 marks)

(c)Let S, T  M such that .

Using (a) and (b) or otherwise, show that

[HKAL92] (5 marks)

e.g.36Eigenvalue and Eigenvector

Let and let x denote a 2x1 matrix.

(a)Find the two real values and of with

such that the matrix equation

(*)

has non-zero solutions.

(b)Let and be non-zero solutions of (*) corresponding to

and respectively. Show that if

and

then the matrix is non-singular.

(c)Using (a) and (b), show that

and hence where n is a positive integer.

Evaluate .[HKAL82]

Prepared by K. F. Ngai

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