# OEF Déterminant --- Introduction ---

Ce module regroupe pour l'instant 26 exercices sur le déterminant d'une matrice carrée.

### det 2x2

Compute the determinant of the matrix

### det 3x3

Compute the determinant of the matrix

### det 4x4

Compute the determinant of the matrix

### det 5x5

Compute the determinant of the matrix

### Determinant and rank

Let and be two matrices ×, such that and . Then

.

(You should put the most relevant reply.)

### Det and trace 2x2

Compute the determinant and the trace of the matrix

### Det and trace 3x3

Compute the determinant and the trace of the matrix

### Det and trace 4x4

Compute the determinant and the trace of the matrix

### Example 2x2

Find a matrix such that:
• All the coefficients are integers 0;
• det(A) 0.

### Example 2x2 II

Find a matrix such that:
• All the coefficients are integers 0 ;
• det(A) 0 ;
•  =  .

### Example 2x2 IIb

Find a matrix such that:
• All the coefficients are integers 0 ;
• det(A) 0 ;
• .

### Example 2x2 III

Find a matrix such that:
• All the coefficients are integers ;
• det(A) 0 ;
• The product  =  .

### Example 2x2 IIIb

Find a matrix such that:
• All the coefficients are integers 0 ;
• det(A) 0 ;
• ;
• .

### Gauss 3x3

Let A be a 3×3 matrix. We make a series of operations on A:
1. Replace number by the sum of times number plus times number .
2. number by .
3. Replace number by the sum of times number plus times number .
4. Exchange number and .
After these operations, the matrix becomes

.

What is the determinant of A ?

### Gauss 3x3 II

Let A be a 3×3 matrix with det(A) = . We make a series of operations on A:
1. Replace number by the sum of times number plus times number .
2. number by .
3. Replace number by the sum of times number plus times number .
4. Exchange number and .
After these operations, we obtain a new matrix B. What is its determinant?

### Given products

Let A and B be two × matrices of real coefficients. Given that

and ,

what are the determinants of A and B ?

### Parametric 3x3

For which value of the parameter t do we have

det  =  ?

### Parametric 4x4

For which value of the parameter t do we have

det  =  ?

### Row permutation 3x3

Consider the following matrices.

Given that det(A) = , what is det(B) ?

### Row permutation 3x3 II

Consider the following matrices.

Given that det(A) = , what is det(B) ?

### Product and inverse 2x2

Let A and B be two × matrices. Given that

,

what is the determinant of A?

### Product and inverse 3x3

Let A and B be two × matrices. Given that

,

what is the determinant of A?

### Product and inverse 4x4

Let A and B be two × matrices. Given that

,

what is the determinant of A?

### Product and inverse 5x5

Let A and B be two × matrices. Given that

,

what is the determinant of A?

Let A b a × matrix with det(A) = . If we replace the number of A by the sum of times number plus times number , we obtain a new matrix B. What is its determinant?

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