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OEF Determinant

OEF Determinant
--- Introduction ---

This module actually contains 26 exercises on the determinant of a
square matrix.

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:

Replace number by the sum of times number plus times number .

number by .

Replace number by the sum of times number plus times number .

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:

Replace number by the sum of times number plus times number .

number by .

Replace number by the sum of times number plus times number .

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?

Addition of rows

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?

Addition of rows II

Let A be a × matrix. If we replace the number of A by the sum of times number plus times number , we obtain a new matrix B. Given that det(B) = , what is the determinant of A ?

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