Matriculation QS015 2014

S.Y.Chuah

July 9, 2014

CHAPTER 4 :

Matrices & Systems of Linear Equations

4.1 Matrices

(a) Identify the different types of matrices.

(b) Perform operations on matrices.

(c) Transpose a matrix and solve related problems.

4.2 Determinant of Matrices

(a) Find the minors and cofactors of a matrix.

(b) Find the determinant of a matrix.

4.3 Inverse of a Matrix

(a) Find the inverse of a non-singular matrix using:

(i) adjoint Matrix; and

(ii) elementary row operations,

4.4 Systems of Linear Equations with Three Variables

(a) Write a system of linear equations in the form AX = B.

(b) Find the unique solution to AX = B using

(i) Inverse Matrix Method

(ii) Elimination Method

1

4.1 Matrices

A matrix is a rectangular array of numbers arranged in rows and columns which is en-closed by a pair of brackets.

If a matrix has m rows and n columns, it is called an m n matrix or a matrix oforder m n.

The general form of a matrix of order m n can be represented as

a11 a12 a13 a14 . . . a1na21 a22 a23 a24 . . . a2na31 a32 a33 a34 . . . a3na41 a42 a43 a44 . . . a4n...

......

......

...am1 am2 am3 am4 . . . amn

EXAMPLE 1Given

A =

6 3 21 0 45 7 2

. Find the elements a13, a22 and a32.

4.1.1 Different types of Matrices

(i) Row MatrixRow matrix is a 1 n matrix. It consists of one row only.Example:

M =[a11 a12 a13 a14 a15

](ii) Column Matrix

Column matrix is a n 1 matrix. It consists of one column only.Example:

M =

a11a21a31a41a51

(iii) Square Matrix

Square matrix is a nn matrix. It consists of equal numbers of rows and columns.

Example:

M =

a11 a12 a13a21 a22 a23a31 a32 a33

(iv) Zero Matrix

Zero matrix is a m n matrix which every entry is zero and the zero matrix isdenoted by 0.Example:

O =

0 0 00 0 00 0 0

O = [0 00 0

]O =

0 00 00 0

(v) Diagonal Matrix

A =

a11 a12 a13 . . . a1ma21 a22 a23 . . . a2ma31 a32 a33 . . . a3m...

......

......

am1 am2 am3 . . . amm

The diagonal entries of this matrix are a11, a22, a33, a44, . . . , amm.A square matrix which non-diagonal entries are all zeroes is called a diagonal matrix.

M =

1 0 00 2 00 0 3

B =

[1 00 2

]

(vi) Identity MatrixIdentity matrix is a diagonal matrix where its diagonal entries are 1 and denotedby I.

M =

[1 00 1

]= I22

(vii) Lower Triangular MatrixLower Triangular Matrix is a square matrix where aij = 0 where i < j.Example:

M =

a11 a12 a13a21 a22 a23a31 a32 a33

=a11 0 0a21 a22 0a31 a32 a33

(viii) Upper Triangular MatrixUpper Triangular Matrix is a square matrix where aij = 0 where i > j.Example:

M =

a11 a12 a13a21 a22 a23a31 a32 a33

=a11 a12 a130 a22 a23

0 0 a33

4.1.2 Operations on Matrices

If we have two matrices, we say that they are equal if they have the same order andtheir corresponding elements are equal. For examples,[

a b cd e f

]=

[u v wx y z

]if and only if a = u, b = v, c = w, d = x, e = y and f = z.

Example:Find the values of x and y if [

2 xy 4

]=

[2 11 4

]Matrix Addition & SubstractionThe sum of two matrices of the same order is a matrix with elements that are thesums of the corresponding elements of the two given matrices.

**Addition and substraction can only be done on matrices with the sameorder. For example, [

a bc d

]+

[w xy z

]=

[(a + w) (b + x)(c + y) (d + z)

][a bc d

][w xy z

]=

[(a w) (b x)(c y) (d z)

]EXAMPLE 2

If A =

2 1 30 4 56 2 1

and B =5 6 34 1 0

2 5 7

, find(i) A + B

(ii) ABEXAMPLE 3

Solve this A =

[2 3 01 2 5

]+

3 31 22 5

.

Multiplication of a Matrix by a numberThe product of a number k and a matrix M , denoted by kM , is a matrix formedby multiplying each element of M by k.

kM = k

[p qr s

]=

[kp kqkr ks

]

For example, given k = 2 and M = 3 1 02 1 3

0 1 2

. Find kM .kM = 2

3 1 02 1 30 1 2

EXAMPLE 4

Find 10

1.30.23.5

Matrix ProductDefinition 1 Product of a Row Matrix and a Column MatrixThe product of a 1 n row matrix and an n 1 column matrix is a 1 1 matrix givenby

[a1 a2 a3 . . . an

]b1b2b3...bn

=[a1b1 + a2b2 + a3b3 + . . . + anbn

]

From the product of row matrix and column matrix above, we can notice that, if thenumber of elements in both matrices are not equal, is the product defined? The answeris NO, hence we must always make sure that the number of elements in the columnof one matrix must be the same as the number of elements in the row of theother matrix as shown below.

For example,

[2 3 12 1 2

] 1 32 01 2

EXAMPLE 5

(A)

2 11 01 2

[1 1 0 12 1 2 0

]

(B)

[1 1 0 12 1 2 0

] 2 11 01 2

(C)

522

[2 3 0]EXAMPLE 6

(A) If A =

[1 32 4

]and B =

[2 51 3

]Show that AB 6= BA

(B) If A =

[2 10 3

], B =

[2 51 3

]and C =

[3 01 4

]Show that (AB)C = A(BC).

EXAMPLE 7

Given that A =

2 5 12 0 35 1 3

, B =2 61 3

2 5

and C =1 4 33 1 1

0 2 2

. Express each ofthe following as a single matrix.

(A) A2

(B) A3

(C) CA

4.1.3 Transpose of Matrix

The transpose of a matrix A is the matrix obtained by interchanging the rows and columnsof matrix A.The transpose of A is denoted by AT .

If A is an m n matrix, then AT is an nm matrix.

Examples:

If A =

[5 4 21 6 8

], then AT =

5 14 62 8

.If B =

3 1 71 4 87 8 5

, then BT =

Properties of matrix transpose

1. (AT )T = A

2. (kA)T = kAT , k is a scalar

3. (AB)T = AT BT

4. (AB)T = BTAT

EXAMPLE 8

A =[2 3

], B =

143

, C = [1 23 5

]andD =

2 2 20 5 31 3 5

Find

(A) AT , BT , CT and DT

(B) (DB)T

4.2 Determinant of Matrices

If A is a square matrix, then the determinant of the matrix A, denoted by |A|, is a realnumber associated with the matrix.

det

[2 35 1

]=

2 35 1 det

1 2 30 5 72 1 6

=

1 2 30 5 72 1 6

Determinant of a 1 1 Matrix

If A = (a11), then |A| = a11.

Example:If A = (4), then |A| = 4.

Determinant of a 2 2 Matrix

If A =

[a11 a12a21 a22

], then |A| = a11a22 a21a12.

Example:

If A =

[5 18 2

], then |A| = (5)(2) (1)(8) = 2.

EXAMPLE 1Determine the value of k if k 3k k + 1

Before looking at the method to calculate the determinant of 3 3 matrices. Lets lookat minors and cofactors of a matrix which are very useful in finding the determinantof a matrix.

4.2.1 Minors and Cofactors of a matrix

MinorsIf A is an n n square matrix, then the minor, denoted by Mij of the element aij is thedeterminant of the (n 1) (n 1) matrix obtained by deleting the ith row and jthcolumn of matrix A.

For example,

A =

a11 a12 a13a21 a22 a23a31 a32 a33

M23 = Minorofa23 =

a11 a12a31 a32 = a11a32 a31a12

M32 =

EXAMPLE 2 If A =

3 1 42 5 21 4 3

, find(A) M11 (B) M23 (C) M32

CofactorsCofactor of aij is denoted by Cij,

Cij = (1)i+j(Mij)Thus, cofactor is nothing more than a signed minor.

If we are given the matrix, a11 a12 a13a21 a22 a23a31 a32 a33

Then, we denote the determinant as

a11 a12 a13a21 a22 a23a31 a32 a33

As an example,

Minor of a23,M23 =

a11 a12a31 a32

Cofactor of a23, C23 = (1)2+3a11 a12a31 a32

= a11 a12a31 a32

EXAMPLE 3

Find the minor and cofactor of 2 and 5 in the determinant2 0 31 6 51 2 0

Solution: Minor of 2,M11=

6 52 0 =

Cofactor of 2, C11= (1)1+1M11 =

NOTE: The sign used in cofactor, (1)i+j, can be determined rather mechanically byusing a checkerboard pattern of + and sign over the determinant, starting with +in theupper left hand corner. The following sign convention can be used to find determinantof 3 3 matrices.

+ + + + +

4.2.2 Determinant of a matrix