Candidates should be able to:

• carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix (Including non-square matrices. Matrices will have at most 3 rows and columns.)
• recall the meaning of the terms ‘singular’ and ‘non-singular’ as applied to square matrices and, for 2 x 2 and 3 x 3 matrices, evaluate determinants and find inverses of non-singular matrices (The notations det M for the determinant of a matrix M, and I for the identity matrix, will be used.)
• understand and use the result, for non-singular matrices, \((AB)^{-1} = B^{-1}A^{-1}\) (Extension to the product of more than two matrices may be required.)
• understand the use of 2 x 2 matrices to represent certain geometric transformations in the x–y plane, in particular
  – understand the relationship between the transformations represented by \(A\) and \(A^{-1}\)
  – recognise that the matrix product \(AB\) represents the transformation that results from the transformation represented by \(B\) followed by the transformation represented by \(A\)
  – recall how the area scale factor of a transformation is related to the determinant of the corresponding matrix
  – find the matrix that represents a given transformation or sequence of transformations
  (Understanding of the terms ‘rotation’, ‘reflection’, ‘enlargement’, ‘stretch’ and ‘shear’ for 2D transformations will be required.
  Other 2D transformations may be included, but no particular knowledge of them is expected.)
• understand the meaning of ‘invariant’ as applied to points and lines in the context of transformations represented by matrices, and solve simple problems involving invariant points and invariant lines. (e.g. to locate the invariant points of the transformation represented by \(\left( \begin{array}{cc} 6 & 5 \\ 2 & 3 \end{array} \right)\), or to find the invariant lines through the origin for \(\left( \begin{array}{cc} 4 & -1 \\ 2 & 1 \end{array} \right)\), or to show that any line with gradient 1 is invariant for \(\left( \begin{array}{cc} 2 & 0 \\ 1 & 1 \end{array} \right)\).

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Matrices – Further Pure Maths 1 Video (4 Parts) (3:02:20)

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