Matrix Rank Worksheet

Solutions

1Calculate the rank of the matrix

Rank of the Matrix

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

Rank of the Matrix

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

Matrix Rank

4Calculate the rank of the matrix by determinants.

Matrix Rank

5Calculate the rank of the matrix by determinants.

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6Calculate the rank of the matrix by determinants.

Matrix Rank

7Calculate the rank of the matrix by determinants.

Matrix Rank


1

Calculate the rank of the matrix.

Matrix Rank

r3 = 2r1

r4 is null

r5 = 2r2 + r1

r(A) = 2.


2

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

Matrix Rank Operations

r1 - 2 r2

Matrix Rank Operations

r3 - 3 r2

Matrix Rank Operations

r3 + 2 r1

Matrix Rank Operations

Therefore r(A) =2.


3

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

Matrix Rank Operations

r2 = r2 − 3r1

r3= r3 − 2r1

Matrix Rank Operations

r(A) = 3.


4

Calculate the rank of the matrix by determinants.

Matrix Rank Operations

|2|=2 ≠0

Matrix Rank Operations

Matrix Rank Operations

Matrix Rank Operations

r(A) = 2


5

Calculate the rank of the matrix by determinants.

Matrix Rank Operations

Matrix Rank Operations

Matrix Rank Operations

r(B) = 4


6

Calculate the rank of the matrix by determinants.

Matrix Rank Operations

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: c5 = −2 · c1 + c2

Matrix Rank Operations

Matrix Rank Operations

r(C) = 2


7

Calculate the rank of the matrix by determinants.

Matrix Rank Operations

c3 = c1 + c2

Matrix Rank Operations

|2|=2≠0

Matrix Rank Operations

Matrix Rank Operations

r(D) = 2