OEF matrices --- Introduction ---

This module actually contains 49 exercises of various styles on matrices.

Example matrix 2x2

Find a matrix $A={pmatrix}a&b c&d{pmatrix} $ such that trace$(A)=$ et $(A)=$, and none of the elements $a,b,c,d$ is zero.

Column and row 2x3

We have a multiplication of matrices What are the values of $x$ and $y$ ?

Column and row 3x3 I

We have a multiplication of matrices What are the values of and ?

Column and row 3x3 II


Determinant and rank

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

.

(You should put the most relevant reply.)


Det & trace 2x2

Compute the determinant and the trace of the matrix

Det & trace 3x3

Compute the determinant and the trace of the matrix

Diagonal multiplication 2x2

Does there exist a diagonal matrix $D$ such that

Left division 2x2

Determine the matrix $A={pmatrix}a&b c&d{pmatrix} $ such that

Right division 2x2

Determine the matrix such that

Equation 2x2

Suppose that a matrix satisfies the equation . Determine the inverse matrix in function of a,b,c,d.

More exactly, each coefficient of must be a polynomial of degree 1 in a,b,c,d.


Formula of entries 2x2

Let C=(cij) the matrix 2×2 whose entries are defined by

cij = .


Formula of entries 3x3

Let the matrix 3×3 whose entries are defined by

.


Formula of entries 3x3 II

Let
= ( )

be a 3×3 matrix whose entries are defined by a linear formula ci,j=f(i,j)=ai+bj+c.

Determine the function .


Given images 2x2

We have a 2×2 matrix , such that

,
. , .

Determine .


Given images 2x3

We have a matrix , such that

,
,
. , , .

Determine .


Given images 3x2

We have a matrix , such that

,
. , .

Determine .


Given images 3x3

We have a 3×3 matrix , such that

,
,
. , , .

Determine .


Given powers 3x3

We have a matrix , with

, .

What is ?


Given products 3x3

We have two matrices and , with

, .

What are and ?


Matrix operations

Consider two matrices

.

Does make sense?
Does make sense?
Does make sense?
Does make sense?
Does make sense?

Min rank A^2

Let A be a matrix ×, of rank . What is the minimum of the rank of the matrix  ?

Multiplication of 3

We have 3 matrices, , , , whose dimensions are as follows.

MatrixABC
Dimension× × ×
Rows
Columns

Give an order of multiplication of these 3 matrices that makes sense.

In this case, what is the dimension of the matrix product? × rows and columns.


Multiplication 2x2

Compute the product of matrices:

Partial multiplication 3x3

We have an equation of multiplication of matrices × as follows, where the question marks represent unknown coefficients.

Step 1. There is only one determinable coefficient in the product matrix. It is .
(Type c11 for for example.) Step 2. The determinable coefficient is = .


Partial multiplication 4x4

We have an equation of multiplication of matrices × as follows, where the question marks represent unknown coefficients.

Step 1. There is only one determinable coefficient in the product matrix. It is .
(Type c11 for for example.) Step 2. The determinable coefficient is = .


Partial multiplication 5x5

We have an equation of multiplication of matrices × as follows, where the question marks represent unknown coefficients.

Step 1. There is only one determinable coefficient in the product matrix. It is .
(Type c11 for for example.) Step 2. The determinable coefficient is = .


Sizes and multiplication

Consider two matrices and , with

, and .

What is the size of ?

Reply: has rows and columns.


Parametric matrix 2x2

Find the values of the parameters $s$ and $t$ such that the matrix verifies .

Parametric matrix 3x3

Find the values of the parameters and such that the matrix
verifies det and trace .

Parametric rank 3x4x1

Consider the following parametrized matrix.

Fill-in: Following the values of the parameter , the rank of A is at least and at most .

The rank is reached when is .


Parametric rank 3x4x2

Consider the following parametrized matrix.

Fill-in: Following the values of the parameters and , the rank of A is at least and at most .

The rank is reached when is is .


Parametric rank 3x4x1

Consider the following parametrized matrix.

Fill-in: Following the values of the parameter , the rank of A is at least and at most .

The rank is reached when is .


Parametric rank 3x5x2

Consider the following parametrized matrix.

Fill-in: Following the values of the parameters and , the rank of A is at least and at most .

The rank is reached when is is .


Parametric rank 4x5x1

Consider the following parametrized matrix.

Fill-in: Following the values of the parameter , the rank of A is at least and at most .

The rank is reached when is .


Parametric rank 4x5x2

Consider the following parametrized matrix.

Fill-in: Following the values of the parameters and , the rank of A is at least and at most .

The rank is reached when is is .


Parametric rank 4x6x1

Consider the following parametrized matrix.

Fill-in: Following the values of the parameter , the rank of A is at least and at most .

The rank is reached when is .


Parametric rank 4x6x2

Consider the following parametrized matrix.

Fill-in: Following the values of the parameters and , the rank of A is at least and at most .

The rank is reached when is is .


Pseudo-inverse 2x2

We have a 2×2 matrix A, with

  .

Please find the inverse matrix of A.


Pseudo-inverse 2x2 II

We have a 2×2 matrix A, with

  .

Please find the inverse matrix of A.


Pseudo-inverse 3x3

We have a 3×3 matrix A, with

  .

Please find the inverse matrix of A.


Quadratic solution 2x2


Rank and multiplication

Let C be a matrix of size ×, of rank . What is the condition on n, in order that there exist a matrix A of size ×n and a matrix B of size n×, such that C=AB ?

Square root 2x2*

Find a matrix such that where the entries must be non-zero integers.

Symmetry of the plane

What is the nature of the plane transformation given by the matrix  ?

Symmetry of the plane II

Among the following matrices, which one corresponds to the of the plane?


Trace of A^2 2x2


Unimodular inverse 3x3

Compute the inverse of the matrix

  .


Unimodular inverse 4x4

Compute the inverse of the matrix

  .

Other exercises on: matrices   determinant   linear algebra  

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