Determinants and Inverses

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Determinants and Inverses

• Consider the weight suspended by wires problem

One (poor) way is to find inverse or reciprocal
of A, A-1. It is defined that A-1A AA-1 I
the identity matrix If we know A-1, then by
pre-multiplying the equation A-1 A T A-1
Y thus T A-1 Y How to find the
inverse - if one exists (just as 1/0 not exists)
? We shall consider square matrices and one
method of finding the inverse - using
determinants and cofactor matrices. Note
determinants are not just used to find
inverses. In fact, determinants are more useful
than inverses.
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Determinants determinant of A det(A) A
a11 a11
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Cofactors for finding Determinants
Det of matrix without ith row and jth column is
defined as Mij.
e.g for 33 matrix
The cofactor of element i,j, Cij, is (-1)ijMij
(-1)ij 1 or -1 Then the determinant of
any nn matrix is defined as a11 C11 a12
C12 ... a1n C1n Determinant of a 22
matrix is a11 C11 a12 C12 a11
(-1)11a22 a12 (-1)12a21 a11 a22 - a12
a21 Determinant of a 33 matrix a11 C11
a12 C12 a13 C13
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Adjoint Matrix of a square matrix A, Adj A
If Cij is the cofactor of aij, then Adj A,
Cji CijT.
then the matrix of cofactors of A is
i.e. the transpose of the above
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Matrix Inverse (poor method)
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Example
The cofactors are
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Properties (A-1)-1 A (AT)-1 (A-1)T (A
B)-1 B-1 A-1 If AT A-1 then matrix A is
an orthogonal matrix.
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Applying Inverses to Example Systems
Suspended Mass
i.e. T1 300N and T2 360N
Check T10.96 - T20.8 288 - 288 0
T10.28 T20.6 84 216 300 ? Good!
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Electronic Circuit
A 18 (-10-15) - 10 (0 --15) 0 (0-1)
-600
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Thus, i1 0.5A, Check 18 i1 10 i2 9
3 12 i2 0.3A -10 i2 15 i3 -3
3 0 and i3 0.2A -i1 i2 i3 -0.5 0.3
0.2 0
Exercise Use matrix inversion to solve
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Stochastic Matrix what was situation in 1990?
By post-multiplying both sides by inverse of
transition matrix
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Note, 1/0.3 is a scalar, and a k b k a b