Determinants
Pre-requisites: Vectors
A matrix is just a two-dimensional array of numbers, arranged like
so:
That was a 2 by 3 matrix; an n by m matrix is a matrix with nm numbers
arranged into a rectangular array with n rows and m columns:
You can add or subrtrct two matrices of the the same size (both n by m) term
wise and multiply or divide any sized matrix by a scalar constant,
just like you can do for vectors.
Matrices are special in the way that they multiply. If you have an n by m
matrix A and a m by p matrix B, then their product is the n by p matrix
AB which has as its (i,j)-th entry the dot product of A's i-th row and
B's j-th column.
Matrices (and determinants) arose
out of efforts to understand systems of linear equations. If you have
the system of linear equations
a x + by = e
c x + d y = f
it can be rewritten into the matrix equation as
[ a b ] [ x ] = [ e ]
[ c d ] [ y ] [ f ]
If we tried to solve this system, we would get answers like
x = (ed - bf)/(ad-bc) and y = (ec-af)/(ad-bc). The denominator
ad-bc of these expressions is called the determinant of the
matrix
[ a b ]
[ c d ]
The determinant of an n by n matrix has many important properties. We just
saw one: the system of linear equations Ax = y is solvable if and only if
det A is non-zero. Here are some others:
- If you take the matrix A and add its i-th row to its j-th row and call
the new matrix B, then det A = det B.
- If you take the matrix A and multiply its i-th row by some constant c
and call the new matrix B, then det B = c det A.
- If you take the matrix A and switch two of its rows and call the new matrix
B, then det B = - det A.
- det I = 1.
- det (AB) = (det A)(det B).
Exercises:
- Given an example of two matrices A and B for which AB is not the same
as BA.
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Last modified 1 July 1997