If the determinant of the matrix is positive, it must be a rotation or a combination of a rotation and a shear.

It the determinant of the matrix is 1 and the columns or rows have length 1 when considered as a vector then it is a rotation. The matrix (in two dimensions) may be writtenWe can find the rotation angle by equating the transformation matrix to this and solving forNote that in general two one equation needs to be solved for each ofandto get the correct answer.

If the determinant of the matrix is negative, then it must be either a reflection or a combination of a reflection and a shear.

It the determinant of the matrix is -1 and the columns or rows have length 1 when considered as a vector then it is a reflection. The matrix (in two dimensions) may be writtenWe can find the anglethat the reflection line makes with the- axis by equating the transformation matrix to this and solving forNote that in general two one equation needs to be solved for each ofandto get the correct answer.

If the matrix scales a vector, it may scale in thedirection or thedirection or both. If it scales in thedirection only, then it must leave anyvalue unchanged, so that if the transformation is represented bythenso thatSinceandare arbitrary, we must haveandSimilarly, if the matrix represents a scaling in thedirection only, thenandOf course we may multiply matrices representingandscalings to obtain a matrix that scales in both directions simultaneously.

The area of any shape transformed is related to the area of the untransformed shape by the determinant of the transformation matrix,Ifandare the original and transformed shapes then

The anglesthe vector makes with the axes are illustrated above.

Ifthen

]]>In this equationis a constant called the eigenvalue.

The procedure for finding eigenvectors and eigenvalues is quite simple. IfthenThis means that the matrixhas zero determinant. We can solveand solve this equation to find values ofIn general several values of %lambda may be found. Each value ofgives value to at least one eigenvector, and different eigenvectors give rise to different eigenvalues.

Example: Find the eigenvalues and eigenvectors for the matrix

We obtain

Solving this givesor

We find the eigenvectorsby solving

For

Henceand an eigenvector is

For

Henceand an eigenvector is

Notice that any vectors of the formsandare eigenvectors. We choose values ofto make the eigenvectors as simple as possible.

]]>Example: Find factors in the determinant of the matrix

The determinant is(1)

Ifthen the matrix becomes

The first two rows are the same and the determinant is zero, sois a factor.

Ifthen the matrix becomes

The first and third rows are the same and the determinant is zero, sois a factor.

Ifthen the matrix becomes

The last two rows are the same and the determinant is zero, sois a factor.

The matrix can have at most three factors (since it has three rows) sois a factor and can differ from the determinant only by a constant factor. In fact, multiplying out the above expression givesThis differs from (1) only by a factor -1 so the determinant can be factorised as

]]>1. Form the adjoint matrix.

2. Permute the signs according to

3. Take the transpose.

4. Find the determinantof the original matrix and divide the matrix by it, which means multiplying the matrix by a factor

Example: Find the inverse of

We find the adjoint matrix by, for each element, crossing out the elements in the same row and column, and finding the determinant of the submatrix left behind. For instannce, take the element 3 in the top left hand corner. Cross out the top row and the first column. The submatrix left behind iswith determinant 0*1-2*2=-4. This goes in place of the 3. The adjoint matrix formed in this way is:

Now permute the signs, which results in those signs labelled with a “–“ in 2. above changing signs.

Take the transpose to obtain

Finally multiply the matrix by the reciprocal of the determinant of the original matrix.

The inverse is

]]>As shown in the diagram above, two planes intersect in a line. We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication – we have to introduce a parameter. If we take the parameter at being one of the coordinates, this usually simplifies the algebra.

Example: Find the equation of intersection of the planesand

We take the parameter asand putThe equations become

(1)

(2)

(1)-2*(2) gives

Substituteinto (2)

]]>A line can be defined in terms of the cross product using the fact that ifis the position vector of a particular fixed point on the line relative to the origin andis the position vector of a general point on the line, thenis parallel to the line, or more precisely, the tangent vector of the line. This means that(1) whereis the tangent vector along the line sine the cross product of parallel vectors is zero.

We can use this to find the vector equation of a line. Suppose we have the line given by vector equationTakeThe tangent vectorWith we have

To show this gives the same line as the vector equation for the line, we can put

as required by (1).

]]>Henceand

Rearranging the first of these givesSubstitute this into the second to obtain

]]>Every transformation represented by a matrix has at least one invariant point – the origin, since if is the matrix representing T,whereindicates the zero vector with every entry equal to 0. Suppose thatis an invariant ofso that

The above equation means thatis an eigenvector ofwith eigenvalue 1. Not all matrices have such eigenvalues, so this is a condition of a transformation having invariant points other than vec 0 . If such a vectorexists, any scalar multiple ofwill also be invariant since

This means that the eigenvector corresponding to an eigenvalue of 1 will define a line every point of which is an invariant point.

Example:

Suppose a transformationis represented byThe eigenvalues are the solutions to

Ifthe eigenvectorsare the solutions to

Henceand an eigenvector isbut this is not invariant, because the eigenvalue is 3 so

Ifthe eigenvectorsare the solutions to

Henceand an eigenvector isbut this is invariant, because the eigenvalue is 1 so

Also, any scalar multiple ofis invariant so in particularwhich defines the lineis invariant.

]]>can be written as

The system can be solved, the solution found by multiplying both sides by the inverse of the matrix,obtaining

This is only possible if the matrix has an inverse. The matrix has an inverse only if the determinant is non – zero, and the matrix only has a non – zero determinant only if the equations are independent, so that none of the equations can be obtained by adding combinations of the other two.

If the equations are independent in the way described, then the solution is unique, and such a unique solution always exists for a square matrix of coefficients made up of independent equations, where the number of equations is equal to the number of variables.

If the number of equations is LESS than the number of variables, and the equations are independent then in general an infinite number of solutions exist. The solutions can be neatly expressed with the introduction of parameters.

If the number of equations is MORE than the number of variables, the equations cannot be independent. We may be able to reduce the number of equations to form an independent set, possibly obtaining a square matrix so we can solve the system or we may not. If we cannot do this then the system has no solution – it is inconsistent.

For example, consider the system

(1)

(2)

(3)

The system is independent, but (2)-(1) givesand (3)-(2) gives– a contradiction. In general though inconsistency results in the same combination of terms being equal to different numbers egand

]]>