Matrices and Determinants



Advanced Level Pure Mathematics

Chapter 8 Matrices and Determinants

8.1 Introduction : Matrix / Matrices 2

8.2 Some Special Matrix 3

8.3 Arithmetrics of Matrices 4

8.4 Inverse of a Square Matrix 16

8.5 Determinants 19

8.6 Properties of Determinants 21

8.7 Inverse of Square Matrix by Determinants 27

8.1 Introduction : Matrix / Matrices

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

[pic]

is called an m(n matrix.

e.g. [pic] is a 2(3 matrix.

e.g. [pic] 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. [pic] is a matrix of order 3(4.

e.g. [pic] is a matrix of order 3.

3. [pic] is called a row matrix or row vector.

4. [pic] is called a column matrix or column vector.

e.g. [pic] is a column vector of order 3(1.

e.g. [pic]is a row vector of order 1(3.

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

6. [pic] is called a square matrix of order n.

And [pic] is called the principal diagonal.

e.g. [pic] is a square matrix of order 2.

7. Notation : [pic]

8.2 Some Special Matrix.

Def.8.1 If all the elements are zero, the matrix is called a zero matrix or null matrix, denoted by [pic].

e.g. [pic] is a 2(2 zero matrix, and denoted by [pic].

Def.8.2 Let [pic] be a square matrix.

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

(ii) If [pic] for all ij, then A is called a upper triangular matrix.

[pic] [pic]

i.e. Lower triangular matrix Upper triangular matrix

e.g. [pic] is a lower triangular matrix.

e.g. [pic] is an upper triangular matrix.

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

e.g. [pic] is a diagonal matrix.

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

e.g. [pic] , [pic]

8.3 Arithmetrics of Matrices.

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

i.e. [pic].

e.g. [pic] [pic].

N.B. [pic] and [pic]

Def.8.6 Let [pic] and [pic].

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

e.g. [pic]

N.B. 1. [pic] is not defined.

2. [pic] is not defined.

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

e.g.1 If [pic] and [pic]. Find -A and A-B.

Thm.8.1 Properties 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.8.8 Scalar Multiplication.

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

i.e. [pic]

e.g. If [pic] ,

then -2A= ; [pic]

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

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

Thm.8.2 Properties 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.8.9 Let [pic]. The transpose of A, denoted by [pic], or [pic] , is defined by

[pic]

e.g. [pic] , then [pic]

e.g. [pic], then [pic]

e.g. [pic], then [pic]

N.B. (1) [pic]

(2) [pic], then [pic]

Thm.8.3 Properties of Transpose.

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

(a) [pic]

(b) [pic]

(c) [pic]

Def.8.11 A square matrix A is called a symmetric matrix iff [pic].

i.e. A is symmetric matrix [pic]

e.g. [pic] is a symmetric matrix.

e.g. [pic] is not a symmetric matrix.

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

i.e. A is skew-symmetric matrix [pic]

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

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

Def.12 Matrix Multiplication.

Let [pic] and [pic]. Then the product AB is defined as the m(p matrix [pic] where

[pic].

i.e. [pic]

e.g.4 Let [pic]. Find AB and BA.

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

N.B. In general, AB ( BA .

i.e. matrix multiplication is not commutative.

Thm.8.4 Properties 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) [pic].

N.B. (1) Since AB ( BA ;

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

(2) [pic].

(3) [pic]

[pic]

e.g. Let [pic]

Then [pic]

[pic][pic]

But A ( O and B ( C,

so [pic].

Def. Powers of matrices

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

[pic] denotes [pic] .

N.B. (1) [pic]

[pic]

[pic]

(2) If [pic], then [pic]

e.g.6 Let [pic], [pic],[pic] and [pic]

Evaluate the following :

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

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

[pic] .

(b) Find a 2x2 matrix [pic] such that

[pic] and [pic].

(c) If [pic], find the values of [pic].

e.g.8 Let [pic]. Prove by mathematical induction that

[pic] [HKAL92] (3 marks)

e.g.9 (a) Let [pic] where [pic].

Prove that [pic] for all positive integers n.

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

e.g.10 (a) Let [pic] 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 [pic], there exists [pic] such that [pic], 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. [pic]), 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 [pic].

(2) If [pic], then

(a) [pic]

(b) [pic].

(3) [pic]

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

Prove that (i) [pic]

(ii) [pic] for n = 3, 4, 5, ... .

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

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

[pic]. (4 marks)

(c) If X and Y are square matrices,

(i) prove that [pic] implies XY = YX ;

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

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

[HKAL90] (8 marks)

8.4 Inverse of a Square Matrix

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

[pic] and [pic] is usually called the multiplicative inverse of b.

(2) If B, C are matrices, then [pic] 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 [pic]

i.e. [pic].

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

i.e. [pic].

e.g.13 Is [pic]?

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. [pic] and [pic] are both non-singular.

i.e. A is non-singular iff [pic] exists.

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

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N.B. (1) [pic], 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.14 Let [pic].

(a) Show that [pic].

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

(c) Find a matrix X such that [pic].

Properties of Inverses

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

(a) [pic].

(b) [pic] is a non-singular and [pic].

(c) [pic] is a non-singular and [pic].

(d) (A is a non-singular and [pic].

(e) AB is a non-singular and [pic].

Proof Refer to Textbook P.228.

8.5 Determinants

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

(a) If n=2, [pic]

(b) If n=3, [pic]

or [pic]

[pic]

e.g.15 Evaluate (a) [pic] (b) [pic]

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

N.B. [pic]

[pic]

or [pic]

or . . . . . . . . .

By using [pic]

e.g.17 Evaluate (a) [pic] (b) [pic]

8.6 Properties of Determinants

(1) [pic] i.e. [pic].

(2) [pic]

[pic]

(3) [pic]

(4) [pic]

(5) If [pic], then [pic]

(6) [pic]

(7) [pic]

[pic]

N.B. (1) [pic]

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

(8) [pic]

N.B. [pic]

e.g.18 Evaluate (a) [pic] , (b) [pic]

e.g.19 Evaluate [pic]

e.g.20 Factorize the determinant

[pic]

e.g.21 Factorize each of the following :

(a) [pic] [HKAL91] (4 marks)

(b) [pic]

Def. Multiplication of Determinants.

Let [pic] , [pic]

Then [pic]

[pic]

Properties :

(1) det(AB)=(detA)(detB) i.e. [pic]

(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.22 Prove that [pic]

Minors and Cofactors

Def. Let [pic], then [pic] , the cofactor of [pic] , is defined by [pic] , [pic] , ... , [pic].

Since [pic]+[pic][pic]

[pic]

Thm. (a) [pic]

(b) [pic]

e.g. [pic], [pic], etc.

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

(a) Prove that [pic]

(b) Hence, deduce that [pic]

8.7 Inverse of Square Matrix by Determinants

Def. The cofactor matrix of A is defined as [pic].

Def. The adjoint matrix of A is defined as

[pic].

e.g.24 If [pic], find adjA.

e.g.25 (a) Let [pic], find adjA.

(b) Let [pic], find adjB.

Thm. For any square matrix A of order n ,

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

| [pic] |

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Thm. Let A be a square matrix. If detA ( 0 , then A is non-singular and [pic].

Proof Let the order of A be n , from the above theorem , [pic]

e.g.26 Given that [pic], find [pic].

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

e.g.28 Given that [pic], find [pic].

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

e.g.29 Show that [pic] is non-singular.

e.g.30 Let [pic], where [pic].

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

(b) If x=3 , find [pic].

N.B. A is singular (non-invertible) iff [pic] 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) [pic]

(2) [pic]

(3) [pic]

(4) [pic] for any positive integer n.

(5) [pic]

(6) The inverse of a matrix is unique.

(7) [pic]

N.B. [pic]

(8) If A is non-singular , then [pic]

[pic]

N.B. [pic]

(9) If A is non-singular , then [pic]

[pic]

(10) [pic] [pic]

(11) If [pic], then [pic].

(12) If [pic], then [pic] where n ( 0 .

e.g.31 Let [pic] , [pic] and [pic].

(a) Find [pic] and [pic].

(b) Show that [pic].

(c) Hence, evaluate [pic].

e.g.32 Let [pic] and [pic].

(a) Find [pic].

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

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

(b) Given that [pic],

use (a) , or otherwise , to show [pic].

Hence deduce that [pic]. [HKAL93] (7 marks)

e.g.34 (a) If ( , ( and ( are the roots of [pic], find a cubic equation whose

roots are [pic].

(b) Solve the equation [pic] .

Hence, or otherwise, solve the equation

[pic]. [HKAL94] (6 marks)

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

define [pic].

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

(i) [pic],

(ii) [pic],

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

(5 marks)

(b) Let A ( M.

(i) Show that [pic],

where I is the 2x2 identity matrix.

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

A is singular and [pic]. (5 marks)

(c) Let S, T ( M such that [pic].

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

[pic] [HKAL92] (5 marks)

e.g.36 Eigenvalue and Eigenvector

Let [pic] and let x denote a 2x1 matrix.

(a) Find the two real values [pic] and [pic] of [pic] with [pic]>[pic]

such that the matrix equation

(*) [pic]

has non-zero solutions.

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

[pic] and [pic] respectively. Show that if

[pic] and [pic]

then the matrix [pic] is non-singular.

(c) Using (a) and (b), show that [pic]

and hence [pic] where n is a positive integer.

Evaluate [pic]. [HKAL82]

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