Linear Algebra Powertool

Finding the Inverse of a Matrix

Worksheet by Russell Blyth

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

Enter matrix to be inverted

> A:=matrix([[1,4,-1],[-2,0,3],[3,6,1]]);

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Augment the matrix by appending the identity

> K:=augment(A,[1,0,0],[0,1,0],[0,0,1]);

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Reduce the augmented matrix to reduced row echelon form

> K:=rref(K);

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If the n x n identity appears in the first n columns of the matrix, the last n x n columns is the inverse; extract it using the submatrix command

> AINV:=submatrix(K,1..3,4..6);

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Check the result:

> evalm(A &* AINV);

> evalm(AINV &* A);

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The linalg package has a built-in inverse command

> inverse(A);

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If the matrix has no inverse, the procedure above will not give the nx n identity in the first n x n columns of K. Here is how the inverse function behaves:

> B:=matrix([[2,4],[4,8]]);

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> inverse(B);

Error, (in inverse) singular matrix

The linalg package includes a routine to generate random matrices with integer entries, for testing purposes. Let's use it to generate matrices and inverses.

> C:=randmatrix(4,4);

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> inverse(C);

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By default, the entries of the randomly generated matrix fall between -99 and 99. An option can be added to specify the allowed range.

> E:=randmatrix(4,4,entries=rand(-5..5));

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(The variable name D is protected, and is used for the differential operator)

> D(x^2);

> inverse(E);

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