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=(c_{ij}) the matrix 2×2 whose entries are defined by c_{ij} = .
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 c_{i,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. Matrix  A  B  C

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.
Fillin: 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.
Fillin: 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.
Fillin: 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.
Fillin: 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.
Fillin: 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.
Fillin: 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.
Fillin: 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.
Fillin: Following the values of the parameters
and
, the rank of A is at least
and at most
.
The rank is reached when
is
is
.
Pseudoinverse 2x2
We have a 2×2 matrix A, with
. Please find the inverse matrix of A.
Pseudoinverse 2x2 II
We have a 2×2 matrix A, with
. Please find the inverse matrix of A.
Pseudoinverse 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 nonzero 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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