Matrices add in the natural way:

Example:

**Multiplying Matrices **

Multiplying is a little more complex. Remember that you multiply rows by columns.

Example:

**Determinants of Matrices **

To find the determinant of a matrix multiply diagonal corners together and subtract.

Example:

** Inverses of Matrices **

Each entry in the matrix can be divided byin the natural way. An example is shown below

]]>From the graph we can a see solution in the intervalis positive in this region since both factors are negative so multiply to give a positive number.

We solve

By inspection the resulting quadratic equation does not factorise hence we use the quadratic formula

For the quadratic

Onlyis in the intervalso this is the root and because we want the the line to be above the curve we choose and

From the graph we can a see solution in the intervalis negative in this region since one factor is negative and the other positive so multiply to give a negative number. We introduce a minus to allow for this.

We solve

By inspection the resulting quadratic equation does not factorise hence we use the quadratic formula

For the quadratic

From the graph onlyis in the interval sosince we want the line to be above the curve.

Hence

]]>Suppose then that we have the matrix

This matrix has positive determinant so one of the transformations involved is a rotation.

The determinant of the matrix isWe can divide each element of the matrix byto give

The determinant of this matrix is 1 and the each column and row is of magnitude one so the matrix is a rotation matrix. We can equate this to the rotation matrix and find the angle of rotation.

Identifying matrix entries in the upper left and lower left positions gives the equations

ThenThe matrix represents an enlarge with scale factorcentre the origin and a rotation anticlockwise by 69.44 degrees.

The order of transformation in this particular case has no effect because the matrix representing the enlargement isand this is a multiple of the identity.

]]>Suppose then that we have the matrix

This matrix has negative determinant so one of the transformations involved is a rreflection.

The determinant of the matrix isWe can divide each element of the matrix byto give

The determinant of this matrix is -1 and the each column and row is of magnitude one so the matrix is a reflection matrix. We can equate this to the reflection matrix and find the angle of rotation.

Identifying matrix entries in the upper left and lower left positions gives the equations

ThenThe matrix represents an enlargement with scale factorcentre the origin and reflection in the line making an angle of 34.1 degrees with the x axis. The order of transformation in this particular case has no effect because the matrix representing the enlargement isand this is a multiple of the identity.

]]>(1)

Suppose that we have the equationwith rootsandthen comparing with (1) givesand

We can use simple algebra to evaluate many expressions given in terms ofandwithout ever knowing what the values ofand

Example:

Example:

Example:

Example:

Example:

Any function symmetric in terms ofandcan be evaluated in this way. Symmetric here means that ifandare interchanged then the expression is unchanged:

This is the third example above.

]]>Ifwith eachreal then ifis a complex number that is a root of the above equation then the complex conjugateis also a root.

Proof

Ifis a root ofthen

Taking complex conjugates of both sides gives

is a root.

This means thatandare both factors hence so isThis expression will have real coefficients we can possibly find expressions of this sort one by one and perform long division ofby these in turn or use some other method to factorise out the quadratics hence factorisinginto quadratics then linear factors.

Example:has a factorUse this to factorise the cubic expression.

The coefficients of the cubic are real, so sinceis a root, so ishenceandare factors. Henceis a factor.

Inspection ofgivesso the other factor isand the cubic expression factorises:

Example: The quintic polynomialhas a factorUse this to factorise the expression.

The coefficients of the cubic are real, so sinceis a root, so ishenceandare factors. Henceis a factor and the quintic factorises into two quadratics.

Inspection ofgivesso the other quadratic expression iswhich has the rootsand

The full factorisation is

]]>In the diagram belowis the curveandis the curveW e have to find the shaded are and this must be done in several stages.

We find the points A and B of intersection of the two curves, specifically, we find thevalues andHaving found these, the shaded area will consist of the sum of two integrals:

We integrate forthe cardioid betweenandand forbetweenand Notice thatis negative.

The pints of intersections are the solutions to

Forwe integrate betweenandor anticlockwise from

The area of the circle between A and B is

The shaded area is the sum of these

]]>The simplest case is when a line is transformed. To find the equation of the lineafter transformation by the matrixwrite line line as the vectorthen

Thenand

Makethe subject of both equations and equate the result to give

Now make y' the subject to giveFinally drop the ' to give

More generally we multiply the matrixby the vectorobtainingandin terms ofandthen solve these equations to findandin terms ofandFinally substitute forandinto the original equation of the curve to obtain an equation relating andFinally drop the ' as in the example above.

Suppose that the curveis rotated byThe matrix representing this rotation isand

Then and

Adding these two equations givesand subtracting them gives

Substituting these into the original equation of the curvegives

Expanding the brackets giveswhich simplifies to

]]>Since the coefficients of the polynomial are real, the purely complex rootsand occur as a complex conjugate pair so that we can writeandIf the real root isthe the polynomial can be written as

Then from considering the coefficients of

(1)

(2)

and the constant term gives(3)

(3) divided by (2) gives

Then from (1)

Then from (2),

Then

]]>Example: Find the equation of the tangent and normal for the curve given in parametric coordinates asat

At

Tangent:

Normal:

If we have the pointbut notthen we have to use the point to find

Example: Find the equation of the tangent and normal for the curve given in parametric coordinates asat the point

We have to find the value ofandFrom these two,

Tangent:

Normal:

]]>