Matrices and Inverses Flashcards
determinant of 2x2 matrix
ad - bc
what is a determinant
area scale factor of a matrix
determinant of 3x3 matrix
a (det minor a) - b (det minor b) + c (det minor c)
minor a i.e a 2x2 matrix ignoring a row and column
negative determinant
opposite orientation
non-singular or singular matrix
determinant not 0 so has an inverse and one point maps to one image
determinant = 0 is singular, no inverse and multiple points map to one image
det(m) x det(n)
= det(mn)
criteria for matrix to have an inverse
non 0 determinant
square matrix
inverse of a 2x2 matrix
1/det A x originial matrix with a and d swapped and sign on b and c changed
what is a matrix inverse
reverses the transformation
such that A x A-1 = I
what extra thing should you do when multiplying matrices on two sides
specify which side multiplication is occuring
inverse of 3x3 matrix
1/ det A x original matrix replaced with det minor of each cell and sign change in even cells, transposed so rows become columns and vice versa
how to use matrices to form simultaneous equations
each row of a matrix should be a line of the simultaneous equation with the right side on the right side and left side is numbers without x y z multiplied by a 3x1 matrix with x y and z alone
(or 2x1 with just x and y)
solving matrix simultaneous equations
x the solution matrix by the inverse of the equation matrix on the left side
times when matrix equations dont work
determinant 0 - either equations are consistent so coefficients can be equated and have the same solution (so infinite solutions)
or equations form a triangular prism with no solutions
or planes are parallel where coefficients can be equated but then have different solutions (no solutions)
only works when all 3 planes meet at one point
