# Matrix Worksheets

### Solutions

1Given the matrices:

Calculate:

**(A + B) ^{2};
(A - B)^{2};
(B)^{3};
A · B^{t} · C.**

2Prove that:** A ^{2} − A − 2 I = 0**, where:

3 Given the matrix A =

4Determine the matrix that is needed to premultiply by the matrix

in order to obtain the matrix .

5Find all matrices that commute with the matrix:

6 Determine the matrices A and B that verify the system:

7 A factory produces two models of washing machines, A and B, in three available finishes: N, L and S. Model A is produced in 400 units in Finish N, 200 units in Finish L and 50 units in Finish S. Model B is produced in 300 units in Finish N, 100 units in Finish L and 30 units in Finish S. Finish N takes 25 hours of workshop time to complete and 1 hour of administration. Finish L takes 30 hours of workshop time and 1.2 hours of administration. Finally, Finish S takes 33 hours of workshop time and 1.3 hours of administration.

1.Represent the information in two matrices.

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

8A furniture company manufactures three different models of shelves: A, B and C. Each of these models is offered in both large and small sizes. Everyday, the factory produces 1,000 large and 8,000 small shelves in Type A, 8,000 large and 6,000 small shelves in Type B and 4,000 large and 6,000 small shelves in Type C. Regardless of the type, each large shelf has 16 screws and 6 supporting brackets while each small shelf has 12 screws and 4 supporting brackets.

1Represent this information in two matrices.

2Find a matrix that represents the quantity of screws and supports brackets necessary for daily production of each of the six model-sized shelves.

9Given the matrices:

Calculate the value of X in the following equations:

## 1

Given the matrices:

Calculate:

**(A + B) ^{2};
(A − B)^{2};
(B)^{3};
A · B^{t} · C**

## 2

Prove that:** A ^{2} − A − 2 I = 0**, where:

## 3

Given the matrix A =

## 4

Determine the matrix that is needed to premultiply by the matrix

in order to obtain the matrix .

## 5

Find all matrices that commute with the matrix:

## 6

Determine the matrices A and B that verify the system:

*Multiply the second equation by −2 *

Add member to member

If the first equation is multiplied by 3 and the members are added, the result is:

## 7

A factory produces two models of washing machines, A and B, in three available finishes: N, L and S. Model A is produced in 400 units in Finish N, 200 units in Finish L and 50 units in Finish S. Model B is produced in 300 units in Finish N, 100 units in Finish L and 30 units in Finish S. Finish N takes 25 hours of workshop time to complete and 1 hour of administration. Finish L takes 30 hours of workshop time and 1.2 hours of administration. Finally, Finish S takes 33 hours of workshop time and 1.3 hours of administration.

1.Represent the information in two matrices.

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

Matrix production:

Rows: Models A and B Columns: Completions N, L, S

Cost matrix in hours:

Rows: Finishes N, L, S Columns: Cost in hours: T, A

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

## 8

A furniture company manufactures three different models of shelves: A, B and C. Each of these models is offered in both large and small sizes. Everyday, the factory produces 1,000 large and 8,000 small shelves in Type A, 8,000 large and 6,000 small shelves in Type B and 4,000 large and 6,000 small shelves in Type C. Regardless of the type, each large shelf has 16 screws and 6 supporting brackets while each small shelf has 12 screws and 4 supporting brackets.

1Represent this information in two matrices.

Rows: Models A, B, C Columns: Types L, S

Matrix elements of the shelves:

Rows: Types L, S Columns: T, S

2Find a matrix that represents the quantity of screws and supports brackets necessary for daily production of each of the six model-sized shelves.

The matrix that expresses the number of screws and supporting brackets for each shelf model:

## 9

Given the matrices:

Calculate the value of X in the following equations: