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 mn numbers arranged in the form
is called an mn matrix.
e.g. is a 23 matrix.
e.g. is a 31 matrix.
2.If a matrix has m rows and n columns, it is said to be order mn.
e.g. is a matrix of order 34.
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 31.
e.g.is a row vector of order 13.
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 22 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 mn 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 mp 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.ABBA 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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