EAS 6410
LAD 1/29/09
Determinants and the inverse of a matrix
The determinant of a square matrix is a readily calculated quantity that gives important
information about the behavior of that matrix. We can briefly summarize a few points
here.
"a
For a 2x2 matrix, A = $
#c
det A = ad " bc
b%
' we can easily calculate its determinant as:
d&
(det A is sometimes written A )
!
"4 3%
' then det A = (4•1)-(7•3) = -17.
#7 1&
So, for a numerical example in which A = $
!
!
Simple enough in the 2x2 case! For larger (say 3x3 or more) square matrices, the method
of co-factors is used:
a1
a2
a3
b1
b2
b3
c1
b
c 2 = a1 2
b3
c3
!
c2
a
" b1 2
c3
a3
c2
a
+ c1 2
c3
a3
b2
b3
A numerical example is:
"5 3 2%
$
'
For A = 1 4 7 and
$
'
$#2 1 4'&
!
det A = 5
!
4 7
1 7
1 4
"3
+2
= (5 # 9) + (3 # ("10)) + (2 # ("7))= 61
1 4
2 4
2 1
MATLAB easily computes determinants (either numerically or with the Symbolic
Toolbox), so all this multiplication, addition and subtraction is not something we usually
! need to spend time doing.
The determinant has a couple of useful properties/applications. The determinant plays an
important role in matrix inverse theory, but is less used in actual computations.
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EAS 6410
LAD 1/29/09
Matrix inverse
If we want to solve the equation ax = b in a normal scalar problem this is obviously:
x = b/a
v
v
In the case of a system of algebraic equations, instead we write Ax = b where the
overbar indicates that a and b are vectors. Then the solution can be obtained by the
method of Gauss-Jordan elimination easily in MATLAB, using the “reverse division”
operator:
!
x = A\b
Alternatively, we can compute the matrix inverse of A, written inv(A) or A-1.
The inverse of a matrix multiplied by itself is the identity matrix
A•A-1 = I
"1 0 0%
$
'
e.g. I = 0 1 0 in the case of a 3x3
$
'
$#0 0 1'&
Then the solution of our system of algebraic equations can be written as A-1•b=x, this
time using !
boldface type to indicate the vector. MATLAB again can compute these
easily, you can type:
D = inv(A) to see the inverse of the matrix A as a new matrix D
and the solution to Ax = b can be obtained by
x = inv(A)*b
or, identically,
x = D*b
Solving Ax=b with the matrix inverse function is equivalent to using the “reverse
division” operator. Either produces the same result. The reverse division method as
implemented in MATLAB is more computationally efficient, and easier to type.
Computing the inverse of a matrix is valuable, and has wide application. There are many
problems in science known as “inverse problems”, and many rely on this concept.
Consequently, efficient and accurate matrix inversion is an important goal of numerical
computing.
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