Matrices

Further Maths 1 : Matrices - Introduction to matrices, matrice multiplication, inverse matrices, simultaneous matrices.
Overveiw of what you need to know to pass the exam.

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Section 1: Introduction to Matrices

1. Check answers carefully :

2. Make sure you are familiar with the matrices for simple transformations :
Know the matrices for :
Reflection in x & y axis and lines y=x & x=y
Rotation through 90 or 180 degrees about origin.

3. Also become familia with matrices for enlargement and two-way stretches :
Numbers are on leading diagonal and zeros on the other positions.

4. Know the general rotation matrix :
Matrix for rotation of O anticlockwisw about origin.
cos O  - sin O
sin O   cos O

5. Remember useful result about the columns of a matrix :
Image point (1,0) gives first column
Image point (0,1) gives second column

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Section 2: Matrix Multiplication

1. Make sure you can do matrix multiplication confidently :

2. Remember matrix multiplication is not commutative :
General AB does not = BA

3. Make sure matrices are in the correct order for composite transformations :
Remember transformation A follwed by B represent BA matrix.

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Section 3: Inverse Matrices

1. Remember rule for the inverse matrix product :
For square matrices M and N, (MN)-1 = N-1 M-1.

2. Make sure you understand the significance of a zero determinant for a matrix transformation :
For matrix with zero determinant all points are mapped onto a straight line.

3. Remeber the physical significance of the determinant :
Determinant of matrix represents the area of scale factor of the associated transformation. Area scale factor is the square of the scale factor.

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Section 4: Matrices and simultaneous equations

1. Be careful when solving matrix equations for which the matrix has no inverse :
Can mean either no solution or infinitely many solutions.

2. Remember that the origin is always an invarient point for a linear transformation :

3. Make sure that you know the difference between a line of invarient points and invarient line :
Invarient point is mapped to itself.
Invarient line is a line of points each mapped to itself.

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