Given a matrixwith non zero entries only on the leading diagonal, it is very easy to find powers ofIf for example,then A matrixis said to be diagonalizable if we can find a matrixsuch thator equivalentlywhereis a diagonal matrix with non zero entries only on the leading diagonal. Of course only square matrices can be diagonalized, […]

## Rotation Matrices in Three Dimensions

The matrix representing a general rotation in three dimensions is This is a complicated matrix but for rotations about one of the coordinate axes it simplifies considerably. A rotation about the– axis fixes We need a 1 in the top left hand corner and zero’s in the first row and column. Taketo give. A rotation […]

## Linear Independence and Systems of Equations

Any linear system of equations can be written in matrix form, where the matrix is the matrix of coefficients. For example the system 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 […]

## Finding the Line of Intersection of Two Planes

>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. […]

## Descriptions of Transformations Represented by Matrices in Two Dimensions

In general a matrix operating on a vector may reflect, rotate, shear it, or a combination of these. We can deduce the nature of the transformation by inspection of the matrix. If the determinant of the matrix is positive, it must be a rotation or a combination of a rotation and a shear. It the […]

## Properties of Matrix Multiplication

When two matricesandare multiplied to produce a third matrixthe entry in the ith row and jth column, labelledcan be considered as a dot product. If the matrixis considered to be made up of row vectorsand the matrixis considered to be made up of column vectorsthen the elementinis the dot product ofwith This view is helpful […]

## Invariant Points

Any transformationrepresented by a matrixwhose entries are numbers is linear. The matrix will send a line to a line. Sometimes it may send points on a line to some other point on the same line, and sometimes it will send a point to itself. In the second case, the point is said to be invariant. […]

## The Vector or Cross Product

The vector or cross product of two vectorsandwrittenis also a vector. It is at right angles to bothandand if the angle betweenandisthenThe vectorcan be calculated using one of the following methods: This can be hard to remember and many people prefer to find the determinant of a matrix, for which the vectors written in ijk […]

## The Angle Between Two Planes

As illustrated in the diagram below, the angle between two planesandis equal to the angle between the normals to the planesand The components of the normal appear in the cartesian equation of the plane, because ifis any point in the plane,is any fixed point and the normal is thenwhereandare the position vectors ofandrespectively. This results […]

## Finding Inverses of 3×3 Matrices

Finding the inverse of a 3×3 matrix is quite involved. The steps are: 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 […]