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Show that a square matrixAis invertible if and only iftA:Ais invertible Find the matrixAif(I+ 2A A 1

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ISAE Application de l’analyse à la géométrie TD9 J.Saab

1. Evaluate the following determinants:

a)

1 i 1 +i

0 1 2i

i 1 0 0

b)

1 1 2

0 3 0

1 4 2

2. Given the matrix

A= 0 BB BB

@

1 2 1 3 4

1 1 0 2 4

2 1 3 1 2

1 0 1 1 3

0 1 1 1 3

1 CC CC A

Use Gauss elimination to transformAin an upper triangular matrix DeducejAj

3. Let

A= 0

@ 1 2 0

1 3 0

0 1 1

1 A

EvaluatejAjand deduce that Ais invertible

Find the inverse of Aby the cofactors method and then by the Gauss-Jordan method deduce from a) the determinant ofA 1 and then calculate directlyjA 1j

4. For what values of a 2 IR the matrixA = 0

@ 1 1 1

1 2 4

1 3 a

1

A 2 M3(IR) is invertible? Find in this case, its inverse .

5. Show that a square matrixAis invertible if and only iftA:Ais invertible Find the matrixAif(I+ 2A) 1=

0

@ 2 5 5

1 1 0

2 4 3

1 A

1

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