Page 153 - Numerical Analysis Using MATLAB and Excel

Summary

α 11 α 21 α 31 …α n1

α 12 α 22 α 32 …α n2
adjA = α 13 α 23 α 33 …α n3
…… … … …
α 1n α 2n α 3n …α nn

A
n
• An square matrix is called singular if detA = 0 ; if detA ≠ 0 , A is called non-singular.
B
A
I
n
•If and B are square matrices such that AB = BA = I , where is the identity matrix, is
A
A
B
called the inverse of , denoted as B = A 1 – , and likewise, is called the inverse of , that is,
A = B 1 –
• If a matrix is non-singular, we can compute its inverse from the relation
A
1
A 1 – = ------------adjA
detA
I
A
• Multiplication of a matrix by its inverse A 1 – produces the identity matrix , that is,
1 –
AA 1 – = I or A A = I
• If and are matrices whose elements are known, is a matrix (a column vector) whose
B
X
A
elements are the unknowns and and are conformable for multiplication, we can use the
A
X
1 –
relation X=A B to solve any set of simultaneous equations that have solutions. We refer to
this method as the inverse matrix method of solution of simultaneous equations.
• The matrix left division operation is defined as X = A \ B ; this is MATLAB’s solution of
1 –
B
A B for the matrix equation A X⋅ = B , where matrix is the same size as matrix .
X
• We can use Microsoft Excel’s MINVERSE (Matrix Inversion) and MMULT (Matrix Multipli-
cation) functions, to solve any set of simultaneous equations that have solutions. However, we
cannot use them to compute matrices that include complex numbers in their elements.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−35
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