A matrix is a mathematical structure that allows organizing data in rows and columns, facilitating the representation and manipulation of information in various fields, such as engineering, physics, economics and computer science. Matrix exercises help us understand how to work with them, from basic operations like addition and multiplication, to more advanced concepts like determining inverses.

Solve the following problems

1

Given the matrices

Calculate the following sums and subtractions:

a

b

Solution

a

 

We add the elements that are in the same position of both matrices:

 

 

b

 

We subtract the elements that are in the same position of both matrices:

 

2

Given the matrices:

Calcular:


a

b

Solution

a

 

We add the elements that are in the same position of both matrices:

 

 

 

b

 

We subtract the elements that are in the same position of both matrices:

 

 

3

Given the matrices

 

 

Verify if holds

Solution

1 We calculate

 

We multiply row by column (dot product) to obtain element

 

 

 

2 We calculate

 

 

We multiply row by column (dot product) to obtain element

 

 

 

3 With the above we verify that

 

4

Given the matrices

 

 

Verify if holds

Solution

1 We calculate

 

We multiply row by column (dot product) to obtain element

 

 

 

2 We calculate

 

 

We multiply row by column (dot product) to obtain element

 

 

 

3 With the above we verify that

 

5

Given the matrices

Calculate:

a

b

Solution

We remember that the transpose of a matrix is obtained by interchanging the rows with the columns

 

a We calculate 

 

 

b We calculate 

 

6

Given the matrices:

Calculate:

a

b

Solution

We remember that the transpose of a matrix is obtained by interchanging the rows with the columns

 

a We calculate 

 

 

b We calculate 

 

7

Given the matrices:

Calculate:

a

b

Solution

a We calculate

 

 

b We calculate 

 

8

Given the matrices

Calculate:

a

b

Solution

a We calculate 

 

 

b We calculate 

 

9

Find for

 

 

and

Solution

1 We calculate 

 

 

2 We calculate 

 

 

3 We notice that the element found in position matches the power of , so we propose for power

 

 

4 Let's see if the proposed formula holds for power

 

 

 

With the above we verify that the proposed formula is valid for any power

10

Prove that , where

Solution

1 We calculate 

 

 

2 We substitute in the left side of the equation and calculate

 

 

Thus, we have proved the requested equality.

11

Prove that , where

Solution

1 We calculate 

 

 

2 We substitute in the left side of the equation and calculate

 

 

Thus, we have proved the requested equality.

12

Calculate the inverse matrix of

 

Solution

1 Construct a matrix of type

 

 

2 Use the Gauss method to transform the left half, , into the identity matrix, and the resulting matrix on the right side will be the inverse matrix .

 

We make

 

 

 

We make

 

 

We make y

 

 

3 The inverse matrix is

13

Calculate the inverse matrix of

 

Solution

1 Construct a matrix of type

 

 

2 Use the Gauss method to transform the left half, , into the identity matrix, and the resulting matrix on the right side will be the inverse matrix .

 

We make

 

 

We make

 

 

We make and

 

 

3 The inverse matrix is

 

14

Calculate the inverse matrix of:

 

Solution

1 Construct a matrix of type

 

 

2 Use the Gauss method to transform the left half, , into the identity matrix, and the resulting matrix on the right side will be the inverse matrix .

 

 

We make

 

 

We make and

 

 

3 The inverse matrix is

15

Obtain matrices and that satisfy the system:

 

Solution

1 We multiply the second equation by

 

 

2 We add member by member and solve for

 

 

 

3 If we multiply the first equation by 3 and add member by member we obtain:

 

 

16

A factory produces two washing machine models, and , in three finishes: and . It produces model units in finish , units in finish and units in finish . It produces model units in finish , units in finish and units in finish . Finish takes workshop hours and administration hour. Finish takes workshop hours and administration hours. Finish takes workshop hours and administration hours.

1 Represent the information in two matrices.

2 Find a matrix that expresses the workshop and administration hours used for each of the models.

Solution

Production matrix:

 

Rows: Models ; Columns: Finishes

 

 

 

Cost matrix in hours:

 

 

Rows: Finishes ; Columns: Cost in hours:

 

 

 

Matrix that expresses the workshop and administration hours for each of the models:

 

 

17

Calculate the rank of the following matrix:

 

Solution

We perform elementary row operations:

 

1 We make 

 

 

2 We make 

 

 

3 We make 

 

 

Thus .

18

Given:

Calculate the value of in the following equations:

1

2

3

4

5

Solution

We solve for variable in each of the equations

 

1

 

 

2

 

 

3

 

 

4

 

 

5

 

19

Solve in matrix form the system:

Solution

1 We write in matrix form

 

2 We solve the equation

 

 

 

3 Thus, the solution is

 

 

20

Solve in matrix form the system:

Solution

1 We write in matrix form

 

 

2 We solve the equation

 

 

 

3 Thus, the solution is

 

 

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.