Core Pure - Matrices + Linear Transformations Flashcards
What is a square vector?
A matrix when the number of rows and columns are the same
What is an identity matrix?
A square matrix with 1’s along the principal diagonal and the rest of the elements as 0
How do you find the determinant of a 2x2 matrix?
How do you find the determinant of a 3x3 matrix?
What is a singular matrix?
A matrix where Det M = 0
How do you find the inverse of a 2x2 matrix?
What is the property of an inverse matrix?
MM⁻¹ = I
How do you find the inverse of a 3x3 matrix?
- Find the determinant of A, Det A
- Form the matrix of minors, M
- From the matrix of minors, form the matrix of cofactors, C
- Find the Transpose of the matrix of cofactors, Cᵀ
- A⁻¹ = (1/DetA) Cᵀ
What is the transpose of a matrix?
The reflection across the main diagonal
If the determinant of a matrix corresponding to 3 simultaneous equations is singular, is it consistent or inconsistent and how many solutions does it have?
If Det A = 0, the simulatanous equations may:
* Be consistnent and have infinite solutions
* Or be inconsistent and no solutions
If the determinant of a matrix corresponding to 3 simultaneous equations is non-singular, is it consistent or inconsistent and how many solutions does it have?
Is is consistant and has 1 unique solution
What is the gemoetrical representation of a matrix corresponding to 3 simultaneous equations when the determinant does NOT equal 0?
What is the gemoetrical representation of a matrix corresponding to 3 simultaneous equations when it is singular and has infinite solutions?
What is the gemoetrical representation of a matrix corresponding to 3 simultaneous equations when it is singular and has no solutions?
How do you determine whether 3 simultaneous equations with det A = 0 are consistent or inconsistent?
Form 2 equations from the 3 equatiosn (set 1 equal to x,y or z and substitute in to other 2), of the 2 formed equations are multiples of each other, it is consistent, if they are not multiples, they are inconsistent
How do you determine whether 3 simultaneous equations are represented gemoetrically by a prism or 2 or more planes being parallel?
- Prism if at least 2 of the original equations are not mulriples of each other
- Parallel if at least 2 of the original equations are multiples of each other
What is a linear transformation?
A transformation where the origin is unmoved
How do you transforma. point using matrices?
Ma = a’
(a’ = a image)
How do you determine the 2x2 matrix for a transformation?
Using a unit square
What is the rotation matrix for a 2d graph?
What is the equation for the image area?
Image area = Det M x Object acrea
What is the enlargement scale factor?
root(determinant)
What linear transformation does this matrix represent?
Enlargement, centre 0, scale factor a
What linear transformation does this matrix represent?
Shear transfmormation, centre 0, scale factor a parallel to the x axis and scale factor b parallel to the y axis
What is the matrix for matrix A followed by matrix B?
BA
What is an invariant point?
A point which does not move under the transformation
What is an invariant line?
a line where any point on the line is mapped to another point which is also on the line. Howevever, individual points are not invariant
What matrix is a reflection in the plane x=0 represented by?
What matrix is a reflection in the plane y=0 represented by?
What matrix is a reflection in the plane z=0 represented by?
What matrix represents a rotation about the x axis?
What matrix represents a rotation about the y axis?
What matrix represents a rotation about the z axis?
What is the plane x = 0?
The y-z plane
What is the plane y=0?
the x-z plane
What is the plane z=0?
the x-y plane
