# Matrix Rank Worksheet

### Solutions

1Calculate the rank of the matrix

2Calculate the rank of the matrix by the gaussian elimination method.

3Calculate the rank of the matrix by the gaussian elimination method.

4Calculate the rank of the matrix by determinants.

5Calculate the rank of the matrix by determinants.

6Calculate the rank of the matrix by determinants.

7Calculate the rank of the matrix by determinants.

## 1

Calculate the rank of the matrix.

**r _{3} = 2r_{1} **

**r _{4} is null**

**r _{5} = 2r_{2} + r_{1} **

r(A) = 2.

## 2

Calculate the rank of the matrix by the gaussian elimination method.

**r _{1} - 2 r_{2}**

**r _{3} - 3 r_{2}**

**r _{3} + 2 r_{1}**

Therefore r(A) =2.

## 3

Calculate the rank of the matrix by the gaussian elimination method.

r_{2} = r_{2} − 3r_{1}

r_{3}= r_{3} − 2r_{1}

r(A) = 3.

## 4

Calculate the rank of the matrix by determinants.

|2|=2 ≠0

r(A) = 2

## 5

Calculate the rank of the matrix by determinants.

r(B) = 4

## 6

Calculate the rank of the matrix by determinants.

Remove the third column as it is zero, the fourth because it is proportional to the first and the fifth because it is the linear combination of the first and second: **c _{5 } = −2 · c_{1} + c_{2}
**

r(C) = 2

## 7

Calculate the rank of the matrix by determinants.

**c _{3 } = c_{1} + c_{2}**

**|2|=2≠0**

r(D) = 2