MATRICES MATRIX ARITHMETIC - TSFX

The School For Excellence 2011 The Essentials – Mathematical Methods – Notes Page 33

MATRICES A matrix is a rectangular collection of numbers enclosed by round or square brackets. The horizontal lines in a matrix are referred to as rows and the vertical lines are known

as columns. A matrix with m rows and n columns is called a nm matrix. Alternatively, we say

that the order of a matrix with m rows and n columns is nm . i.e. The first number in the order indicates the number of rows. For example: The matrix 931 is a 31 matrix. Two matrices are equal if they are of the same order and if all their corresponding

elements are equal.

For example: If

921

9421

x then 9x .

MATRIX ARITHMETIC Like normal numbers, matrices may be added, subtracted, multiplied and divided.

ADDITION AND SUBTRACTION OF MATRICES Matrices can only be added and/or subtracted if they are of the same dimensions. Addition of matrices is performed by adding corresponding elements.

hdgcfbea

hgfe

dcba

Subtraction of matrices is performed by subtracting corresponding elements.

a b e f a e b fc d g h c g d h

Matrix addition is commutative: ABBA Matrix addition is associative: CBACBA )()( In addition: A O O A A ( ) ( )A A A A O Matrix subtraction is not commutative: ABBA Matrix subtraction is not associative: ( ) ( )A B C A B C


MULTIPLICATION BY A SCALAR The term scalar is used to represent a real number. When a matrix is multiplied by a scalar, we multiply each element in the matrix by that scalar.

dkckbkak

dcba

k

Note: The matrix order is unchanged.

Scalar multiplication is distributive: )( BAkkBkA

Scalar multiplication is associative: AckcAkA )(

Note further that: )()( cAkAkc For example:

If

0421

A ,

5143

B and

2101

C then

)(25 CBA

2101

5143

20421

5

7244

2020105

14488

020105

141623

The School For Excellence 2011 The Essentials – Mathematical Methods – Notes 5

MATRIX MULTIPLICATION Matrix multiplication is possible only if the number of columns in the first matrix is the same as the number of rows in the second matrix. The size of the resultant matrix (the product) is equal to the number of rows in the first matrix and the number of columns in the second matrix.

)())(( pmpnnm When multiplying matrices, we multiply the rows of the first matrix by the columns of the second matrix. For Example:

For Example:

For example: Given that

214312

A and

205213

B

AB

)2(2)5(1)1(4)0(2)2(1)3(4

)2(3)5(1)1(2)0(3)2(1)3(2

131434

Note: A is a (2 3) matrix, B is a (3 2) matrix, and therefore, the product is a (2 2) matrix.


In general: Matrix multiplication is not commutative: BAAB Matrix multiplication is associative: )()( BCACAB

BCACCBA )( CBCABAC )( ACABCBA )( AO OA O

THE IDENTITY MATRIX

In general, matrix multiplication is not commutative i.e. BAAB . There is, however, one exception, that being when a matrix is multiplied by its identity matrix. The identity matrix is a matrix where the main diagonal contains the number 1 for all of its elements, and the other elements are 0 .

For example: I

1001

, I

100010001

The identity matrix is only defined for a square matrix and its order is mm . For example, the above identity matrices are of the order 22 and 33 respectively. Just as the product of a number and 1 is equal to the number, the product of a square matrix and its identity matrix is the original square matrix.

When a matrix A and its identity matrix are multiplied, the original matrix ( A ) is obtained.

AIAAI


THE MULTIPLICATIVE INVERSE (INVERSE MATRIX)

When solving algebraic expressions of the form 62 x , we divide both sides by 2 to solve for x .

The same principles can be applied when solving matrix equations that require a matrix that is part of a product to be cancelled.

For Example:

3179

4231

X

We use the multiplicative inverse or inverse under multiplication under these circumstances.

The multiplicative inverse of A is denoted as 1A and has the property that:

111 AAAA

Given

dcba

the inverse matrix of a 22 is defined as:

bcadA

11

acbd

where bcad is the determinant of A

For example: Given

3275

B , det 5(3) 7(2) 1B

Note:

The determinant may be written as Adet , A or a bc d

.

The Determinant of A is a scalar quantity (a real number).

If the determinant is zero )0( bcad the inverse of the matrix does not exist. Such a matrix is said to be singular and has no inverse.

For example: 3t t

Bt

is singular when 2 3 0t t i.e. When t 0, 3 .

A matrix is said to be regular if its inverse exists.

Only square matrices can have inverses.


FINDING INVERSE MATRICES

To Find an Inverse Matrix of the order 22 without using technology:

Given

dcba

the inverse matrix bcad

A

11

acbd

where bcad is the determinant of A Step 1: Interchange the elements on the leading diagonal. Step 2: Change the sign of the other elements in the matrix.

Step 3: Pre-multiply the matrix withbcad

1.

For example:

Given

4321

C , find the inverse matrix 1C .

Inverse will exist if 0det A : 2)3(2)4(1det A , therefore, 1A does exist.

21

23

12

1324

211C

Use technology to find the inverse of higher order matrices.


SOLVING MATRIX EQUATIONS Just as algebraic expressions can be simplified and solved in the real number system, matrix expressions can also be solved in the same manner. When solving matrix equations, it is important to remember the following differences between matrix algebra and real algebra: • The commutative law for multiplication is not obeyed by matrices.

• It is not always possible to find the multiplicative inverse of a matrix. Examples of common equations to be solved: Let A denote the known matrix. Let X denote an unknown matrix (the matrix we are solving for). (a) Given BAX :

Pre-multiply both sides by 1A ( 1A A I - which isolates X ).

BAAXA 11 1IX A B BAX 1

For example: Find matrix X such that 3 5 0 54 5 10 0

X

.

53

54

11 3 54 5

X

53

54

11

01050

10 56 4

X

(b) Given BXA :

Post-multiply both sides by 1A

11 BAXAA 1XI BA as 1AA I

1 BAX

For example: Find matrix X such that 4 3 1 36 5 2 1

X

.

4 3 2 1.5 1 3 2 1.56 5 3 2.5 2 1 3 2.5

X

6.5 4.5

2 1X


SOLVING LINEAR SIMULTANEOUS EQUATIONS IN TERMS OF TWO VARIABLES

The following set of equations may be written in the given matrix form:

ebyax

fdycx

Matrix Form: a b x ec d y f

For Example: The equation set: 125 yx

232 yx

Is written as 5 2 12 3 2

xy

The matrix a bc d

is often referred to as the coefficient matrix.

The matrix

fe

is known as the solution matrix.

To solve a pair of simultaneous equations of the form BAX :

Pre-multiply both sides by 1A ( 1A A I - which isolates X ).

BAAXA 11 BAX 1

Important Notes: If the equations are not given in the form of ebyax and fdycx , equations

will first need to be transposed so that x and y are on one side of the equation and the constants are on the other side.

If a variable is missing, represent this variable with the number 0 . For example: 5 6x is written as 5 0 6x y .


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SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS Not all simultaneous equations have solutions. When solving linear equations in two unknowns, 3 scenarios may arise: The curves intersect. In this case, a unique solution is obtained.

For unique solution det 0 On a graph the lines intersect at 1 point.

The graphs are parallel and will never intersect. In this case, no solution will be obtained.

We refer to equations that yield no solution as inconsistent equations.

Note: The determinant of the matrix set for such equations is 0 . This is a practical example of singular matrix.

For no solution det 0 or SINGULAR On a graph, the lines are parallel.

The graphs are identical i.e. the curves coincide. No unique solution will be obtained (i.e. an infinite number of solutions is obtained).

We refer to equations that yield infinite solution as dependent equations.

Note: The determinant of the matrix set for such equations is 0 . This is another practical example of a singular matrix.

For no unique solution det 0 or SINGULAR

On a graph, the lines are the same. (Think of this situation as one straight line lying on top of the other).


SIMULTANEOUS EQUATIONS WITH MORE THAN TWO VARIABLES

Consider the following sets of equations: ax by cz j

dx ey fz k

gx hy iz l These equations may be written in matrix form as illustrated below:

a b c x jd e f y kg h i z l

To find the solution to this matrix equation, we pre-multiply both sides of the equation with

the inverse of matrix

a b cd e fg h i

. If this 3 3 matrix is denoted as A , its inverse is written

as 1A and

xX y

z

,

jB k

l

then”

AX B 1 1A AX A B

IX 1X A B

The same approach is used to solve 5 equations in terms of up to 5 unknowns. Note: As students are not required to find the inverses of matrices other than 2 2 , simultaneous equations in terms of 3 or more unknowns must be solved using the CAS. For example: Solve the equations 0x y z

2 3 1x y z

2 2 7x y z

Let

1 1 12 3 12 1 2

A

,

xX y

z

,

017

B

1

112

X A B


GEOMETRIC REPRESENTATION OF SOLUTIONS TO EQUATIONS IN TERMS OF 3 UNKNOWNS

A linear equation in terms of 3 unknowns defines a plane. The intersection of 3 planes is either: (a) a point

(b) a line

(c) or there is no common intersection at all.


SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS WITHOUT UNIQUE SOLUTIONS

If the matrix obtained from the coefficients of a system of simultaneous linear equations is singular (i.e. It has determinant=0), then we have one of two situations: 1. There are no solutions. (a) If we are solving for two variables ( , )x y then the lines described by the two equations are parallel and not touching. (b) If we are solving for three variables ( , , )x y z then two of the three planes described by the equations are parallel and not touching. 2. There are an infinite number of solutions. (a) For two variables, both equations describe the same line: One equation is an exact multiple of the other. (b) For three variables, two of the three equations describe the same plane: Two of the three equations are exact multiples of each other. In these cases, we describe the solution through the use of a real number parameter such as . We designate one of the variables to be equal this parameter (eg. Let z ) and writing the other variables (eg. ,x y ) in terms of . For example: 2 (1)x y and 2 2 4 (2)x y As 2 (1) (2) , both equations describe the same line. Let ,y R Equation (1) gives: 2x Therefore, the lines can be described by {(2 , ) : }R .

PARAMETERS ON THE TI89-T: If we try to solve questions of this type on the TI89-T using the SOLVE and and commands, the answer is given in terms of the parameter where @ indicates the use of a parameter and n how many parameters have been used in this session with the calculator.


ALGEBRA & INDICIAL/EXPONENTIAL FUNCTIONS An indicial or exponential expression is an algebraic expression involving a number that is raised to a power.

For Example: 53 is an indicial expression.

The general form for indicial expressions is written as na x , where: x represents the index or power or exponent. a represents the base. n represents the basic numeral.

INDEX LAWS

Indicial expressions may be simplified by applying the following Index Laws.

MULTIPLYING INDICIAL EXPRESSIONS

To multiply indicial expressions with the same base, add the powers.

nmnm aaa

For Example: 16376376 aaaaa

DIVIDING INDICIAL EXPRESSIONS

To divide indicial expressions with the same base, subtract the powers.

nmnm aaa

For Example: 4373

7

993

27 xxxx

REMOVING BRACKETS

To remove brackets around expressions, multiply the power on each number and pronumeral to the power on the brackets.

mnnm aa

For Example: 9633323332 822 yxyxyx


ZERO POWERS

Any term or number (except for 0), that has been raised to the power of zero is equal to 1.

10 a

For Example: 12 0863 zyx

NEGATIVE POWERS

When a term is moved from the numerator to the denominator or vice versa, the sign of the power changes.

nn aa

1, 0a

Indicial expressions with negative powers may therefore be written with positive powers by moving terms from the numerator to the denominator, and visa versa.

For Example: 44

141 33

3

aaa

For Example: 77272

91

313

bbb

RATIONAL EXPONENTS

In general: qq aa 1 and p

qpqqp

aaa

1

Therefore, 1

2a a and 1 33a a

For example: 251

51

125

1

125

1125 22332

32


SIMPLIFYING INDICIAL EXPRESSIONS METHOD: Step 1: Remove brackets. Step 2: Apply the appropriate Index Laws. Step 3: Re-write the answer with positive powers.

To simplify indicial expressions, it may be necessary to apply more than one index law.

WATCHOUTS You can only use index laws when multiplying or dividing indices with the same

base.

Terms separated by addition or subtraction need to be simplified by applying index laws in reverse.

For example: As nmnm aaa then m n m na a a

11 222 nn

Terms with different bases cannot be simplified by applying index laws without applying the “change of base rule”.

For example: 2 3x y No Solution

For example: 2 2 22 4 2 (2 ) 2 2 2x y x y x y x y

00 0

01 1

10 e

Anything0 = 1. Therefore, if you see the number 1 - think Anything0

xe xe log

623)( xx NOT 5x

3 10 30x x


SOLVING INDICIAL EQUATIONS

Recall that the process of solving equations in the form: 0expression locates the X intercepts on the corresponding graph.

1 2expression expression locates the points of intersection of the graphs

1y=expression and 2y expression .

There are three different approaches that may be applied to solve indicial expressions: Equating powers. Raising expressions to reciprocal powers. Applying logarithms. Which method is to be applied depends upon the manner in which the equation is presented, and whether the unknown is the basic numeral, base or power. METHOD: Step 1: Apply the appropriate index laws to reduce the given equation to as few terms as possible. Step 2: Solve the equation writing your answer with positive powers.


SOLUTION STRATEGIES

IF THE EQUATION CAN BE REDUCED TO 2 TERMS Write each term on either side of the equality sign and solve. You will typically obtain one of the following equation formats: (a) Indicial expressions where the power is unknown. Manipulate the given equation so that the bases are the same, and then equate

powers. i.e. If nm aa then nm .

For example: Solve 813 1 n

41 33 n 41 n 5n

(b) Indicial expressions where the base is unknown.

If the unknown is the base, raise both sides of the indicial equation to the reciprocal of

the power of the existing index. i.e. xxxab

ba

1 .

For example: Solve for x given that 523

x .

323

2

23

5

x 3

2

5 x

Note: If the power on the base is even (eg. 6x ) OR if the numerator in the power is

even (eg. 43x ), the solutions for x will be positive or negative.

(c) If the above approaches are not possible:

Take 10log or elog of both sides of the equation or solve using technology (this will be

a necessity for questions involving combined functions).

For example: 5.03 x

5.0log3log 1010 x

10 10log 3 log 0.5x

10

10

log 0.5 0.631log 3

x


OTHER SOLUTION STRATEGIES (a) Apply the Null Factor Law:

Bring all terms to one side of the equation, factorise and apply the Null Factor Law.

For example: 036 xx ee

3 3( 1) 0x xe e

0xe or 3 1xe

3 0xe e 0x

Note that 0xe has no solution.

(b) Assess whether the equation is a disguised quadratic expression. Is one power twice the value of another power? If yes, the expression could be a disguised quadratic.

For example: Solve 22 5(2 ) 4 0x x

2(2 ) 5(2 ) 4 0x x

2 5 4 0A A where 2xA

For example: Solve 8 42 2 4 0x x

4 2 4(2 ) (2 ) 4 0x x

2 4 0A A where 42 xA

SUMMARY OF SOLUTION PROCESS Step 1: Can I make the bases the same?

If yes, use the following rule: If nm aa then nm . Step 2: Is the unknown the base?

If yes, use xxxab

ba

1 .

Step 3: Can I take log10 or loge of both sides of the equation?


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WATCHOUTS There are no values of the power x for which an exponential

function without translations is less than or equal to zero

i.e. 0xa and xa ve .

For example: 05 x can’t be solved. Answer is undefined.

numbervex 5

If terms are separated by multiplication or division, check to see if the bases are the same. If yes, simplify expressions using Index Laws.

Powers consisting of two or more terms may be simplified by writing the equation as a

series of terms which contain just one term in the power. Use the Index Laws in reverse

to achieve this i.e. nmnm aaa . For example: 11 222 nn If terms are separated by addition and subtraction YOU CANNOT apply Index Laws

directly. Use Index Laws to write each term as products and remove common factors

For example: nnnnnn 25.1)5.02(2222222 1111 Given an exponential equation with two exponential terms – and one term is double the

power of the other – the equation may be a disguised quadratic. Use Let A = method.

12 tt ee 12t

t

ee

t

ttt

eeee 2. 02

2 tt ee , 022 AA etc

Always check answers by substitution or by finding the points of intersection using your calculator.

Always test the validity of your solutions by substituting the values of x into the given

equation. If asked to solve an inequality - proceed using an equality sign. Once solutions have

been obtained, sketch a graph and use logic to determine the appropriate answer. Test a point before leaving the question.

When an inequality is multiplied or divided by a negative number, the direction of the

inequality must be changed.

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