Matrices

• What is a matrix?

  A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is the same as a set of coordinates
• How are matrices classified?

  By their size (order)

  number of rows x number of columns
• Order of a matrix= the dimensions of the matrix
• Matrix with n rows and m columns is called an mxn matrix
• A 1x1 matrix is called a scalar and works as a normal number
• If 2 matrices have the same order (1x1 / 1x1) we can add them together by adding the elements together
• To multiply two matrices they must be compatible - the number of columns in the first matrix must equal to the number of rows in the second matrix
• You can multiply a matrix by a scalar by multiplying every element in the matrix by the scalar
• Scalar Multiplication of Matrices

  Multiplying a matrix by a number (scalar) distributes the multiplication to each element of the matrix.
• Matrix Multiplication

  Multiply two matrices A and B, if the number of columns in A is equal to the number of rows in B.
• Matrix Transpose

  Switching the rows and columns of a matrix.
• You CANNOT multiply a column matrix by a row matrix
• You can only multiply a row matrix and a column matrix
• What is an identity matrix?

  [10]

  [01]

  A identity matrix means if you multiply it by the matrix by the identity it is equivalent to multiplying it by 1
• Translation Matrix

  A transformation that moves a matrix by a certain displacement.
• Rotation Matrix

  A transformation that rotates a matrix around the origin by a certain angle.

  [0-1] rotation around (0,0) through 90* anti-clockwise

  [1 0]

  -

  [0 1] rotation around (0,0) through 90* clockwise

  [-10]
• Scaling/ Enlargement Matrix


  A transformation that increases or decreases the magnitude of a matrix.

  [k 0] Enlargement centre (0,0) scale factor of k

  [0 k]
• Reflection Matrix


  A transformation that flips a matrix over a certain line or plane.

  Reflection in x axis:

  [1 0]

  [0-1]

  Reflection in the y axis

  [-1 0]

  [0 1]

  Reflection in x=y

  [0-1]

  [-1 0]

  Reflection in y=x

  [0 1]

  [1 0]
• Shear Matrix

  A transformation that changes the shape of a matrix by moving one or more of its corners.
• Stretch matrix

  Stretch scale factor x k in y direction:

  [1 0]

  [0 k]