linear algebra As you mav have noted, computing inverses often produces some nas
ID: 2970620 • Letter: L
Question
linear algebra
As you mav have noted, computing inverses often produces some nasty fractions However, some matrices have inverses that involve no fractions. For example, from the formula for the inverse of a 2 times 2 matrix given at the beginning of Section 4.1, Find a 3 times 3 matrix A with all nonzero entries such that both A and A-1 have only integral entries. Note that we have not asked for A-1. [Hint: Compute the determinant of the above matrix. How is this relevant?] Let A be an n times n matrix that has only integers as entries. State a necessary and sufficient condition on the determinant of such a matrix that guarantees that the inverse has only integers as entries. Prove your condition. [Hint: Consider the property AA-1 = I.]Explanation / Answer
10) The exercise doesn't say we can't take the identity matrix , so we can take the matrix :
1 0 0
0 1 0
0 0 1
Which is its own inverse and has determinant 1.
11)
det(A^(-1))=1/det(A) from AA^(-1)=1 and more generally remember that A^(-1) = 1/det(A)* Adj(A) where Adj(A) is the adjoint matrix.If det(A)=+-1 and A has only integers values, then A^(-1) will have only integers values, since the terms of the adjoint matrix consist of determinants of integers elements from A and 1/det(A)=+-1
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