Inverting Matrices
Determinants and Matrix
Multiplication
Determinants
• Square matrices have determinants, which
are useful in other matrix operations,
especially inversion.
a11 a12 
• For a second-order square
a a 
matrix, A,
 21 22 
the determinant is
A  a11  a22  a12  a21
Consider the following bivariate
raw data matrix
Subject # 1
2
3
4
5
X
12 18 32 44 49
Y
1
3
2
4
5
from which the following XY variance-covariance matrix is
obtained:
X
Y
X
256
21.5
Y
21.5
2.5
COVXY
r 

S X SY
21.5
256 2.5
 0.9
A  256(2.5)  21.5(21.5)  177.75
Think of the variance-covariance matrix as containing
information about the two variables – the more variable X
and Y are, the more information you have.
Any redundancy between X and Y reduces the total amount
of information you have -- to the extent that you have
covariance between X and Y, you have less total information.
Generalized Variance
• The determinant tells you how much
information the matrix has about the
variance in the variables – the generalized
variance,
• after removing redundancy among
variables.
• We took the product of the variances and
then subtracted the product of the
covariances (redundancy).
Imagine a Rectangle
• Its width represents information on X
• Its height represents information on Y
• X is perpendicular to Y (orthogonal), thus
rXY = 0.
• The area of the rectangle represents the
total information on X and Y.
• With covariance = 0, the determinant = the
product of the two variances minus 0.
Imagine a Parallelogram
• Allowing X and Y to be correlated with one
another moves the angle between height and
width away from 90 degrees.
• As the angle moves further and further away
from 90 degrees, the area of the
parallelogram is also reduced.
• Eventually to zero (when X and Y are
perfectly correlated).
• See the Generalized Variance video clip in
BlackBoard.
Consider This Data Matrix
Subject # 1
X
10
Y
1
2
20
2
3
30
3
4
40
4
5
50
5
Variance-Covariance Matrix
X
Y
COVXY
r

SX SY
X
250
25
25
1
250 2.5
Y
25
2.5
A  250(2.5)  25(25)  0
Since X and Y are perfectly correlated, the generalized
variance is nil.
Identity Matrix
• An identity matrix has 1’s on its main
diagonal, 0’s elsewhere.
1 0 0
0 1 0


0 0 1
Inversion
• The inverted matrix is that which when
multiplied by A yields the identity matrix.
That is, AA1 = A1A = I.
• With scalars, multiplication by
1
a   1.
the inverse yields the scalar
a
identity.
• Multiplication by an inverse
1 a
a  .
is like division with scalars.
b
b
Inverting a 2x2 Matrix
• For our original variance/covariance matrix:
A
1
2
2.5 - 21.5 
1 a22 - a12 
1

*




A 2 - a21 a11  177.75 - 21.5 256
Multiplying a Scalar by a Matrix
• Simply multiply each matrix element by the
scalar (1/177.75 in this case).
• The resulting inverse matrix is:
 .014064698
A 
 - .120956399
1
- .120956399 

1.440225035 
AA1 = A1A = I
a b  w x  row 1  col1 row 1  col2 
c d    y z   row  col row  col  

 
 
2
1
2
2
256 21.5  .014064698
21.5 2.5   - .120956399

 
aw  by ax  bz 
cw  dy cx  dz 


- .120956399  1 0


1.440225035  0 1
The Determinant of a Third-Order
Square Matrix
a11 a12 a13 


a
a
a

A
21
22
23
3


a31 a32 a33 
 a11  a22  a33  a12  a23  a31  a13  a32 
a21  a31  a22  a13  a11  a32  a23  a12  a21  a33
Matrix Multiplication for a 3 x 3
a b c  r s t 
d e f   u v w  

 

g h i   x y z 
ar  bu  cx as  bv  cy at  bw  cz 
dr  eu  fx ds  ev  fy dt  ew  fz 


gr  hu  ix gs  hv  iy gt  hw  iz 
row1  col1 row1  col2 row1  col3 


row 2  col1 row 2  col2 row 2  col3 
row 3  col1 row 3  col2 row 3  col3 
SAS Will Do It For You
•
•
•
•
•
•
•
•
•
Proc IML;
reset print; display each matrix when created
XY ={
enter the matrix XY
256 21.5, comma at end of row
21.5 2.5}; matrix within { }
determinant = det(XY); find determinant
inverse = inv(XY);
find inverse
identity = XY*inverse;
multiply by inverse
quit;
XY
2 rows
256
21.5
DETERMINANT
INVERSE
IDENTITY
2 cols
21.5
2.5
1 row
177.75
2 rows
0.0140647
-0.1209560
2 rows
1
-2.08E-17
1 col
2 cols
-0.120956
1.440225
2 cols
-2.22E-16
1