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Matrix Inverse

MissFlash 2007. 1. 7. 18:47
Matrix Inverse
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The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A^(-1) such that

AA^(-1)==I, (1)

where I is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation A^_ to denote the inverse matrix.

A square matrix A has an inverse iff the determinant |A|!=0 (Lipschutz 1991, p. 45). A matrix possessing an inverse is called nonsingular, or invertible.

The matrix inverse of a square matrix m may be taken in Mathematica using the function Inverse[m].

For a 2x2 matrix

A=[a b; c d], (2)

the matrix inverse is

A^(-1) = 1/(|A|)[d -b; -c a] (3)
= 1/(ad-bc)[d -b; -c a]. (4)

For a 3x3 matrix

A=[a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)], (5)

the matrix inverse is

A^(-1)==1/(|A|)[|a_(22) a_(23); a_(32) a_(33)| |a_(13) a_(12); a_(33) a_(32)| |a_(12) a_(13); a_(22) a_(23)|; ; |a_(23) a_(21); a_(33) a_(31)| |a_(11) a_(13); a_(31) a_(33)| |a_(13) a_(11); a_(23) a_(21)|; ; |a_(21) a_(22); a_(31) a_(32)| |a_(12) a_(11); a_(32) a_(31)| |a_(11) a_(12); a_(21) a_(22)|].

determinant of A is , |a_1 a_2 a_3; b_1 b_2 b_3; c_1 c_2 c_3| ==a_1b_2c_3-a_1b_3c_2-a_2b_1c_3+a_2b_3c_1+a_3b_1c_2-a_3b_2c_1.

 (6)

A general nxn matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.

The inverse of a product AB of matrices A and B can be expressed in terms of A^(-1) and B^(-1). Let

C=AB. (7)

Then

B==A^(-1)AB==A^(-1)C (8)

and

A==ABB^(-1)==CB^(-1). (9)

Therefore,

C==AB==(CB^(-1))(A^(-1)C)==CB^(-1)A^(-1)C, (10)

so

CB^(-1)A^(-1)==I, (11)

where I is the identity matrix, and

B^(-1)A^(-1)==C^(-1)==(AB)^(-1). (12)

SEE ALSO: Gauss-Jordan Elimination, Gaussian Elimination, LU Decomposition, Matrix, Matrix 1-Inverse, Matrix Addition, Matrix Multiplication, Moore-Penrose Matrix Inverse, Nonsingular Matrix, Pseudoinverse, Singular Matrix, Strassen Formulas. [Pages Linking Here]