Matrix Rank and Row Echelon Form

The video tutorial explains the concept of matrix rank, which is the number of linearly independent rows or columns in a matrix. It is always positive except for the zero matrix, which has a rank of zero. The tutorial provides five examples to illustrate how to calculate the rank of different matrices, using row Echelon form and examining linear independence among rows and columns. Each example demonstrates a different approach to finding the rank, highlighting the importance of non-zero rows and linear combinations.

What is the rank of a matrix?

• The number of linearly independent rows or columns

• The number of columns in the matrix

• The number of rows in the matrix

• The number of zero rows in the matrix

Which matrix has a rank of zero?

• A matrix with more columns than rows

• A matrix with more rows than columns

• A zero matrix

• A matrix with all non-zero elements

How do you determine the rank of matrix A?

• By counting the total number of elements

• By finding the number of non-zero rows in its row Echelon form

• By counting the number of columns

• By adding all the elements

What is the rank of matrix A after converting to row Echelon form?

• 3

• 4

• 1

• 2

What is the rank of matrix B?

• 4

• 3

• 2

• 1

How many non-zero rows does matrix B have in its row Echelon form?

• 1

• 2

• 4

• 3

Why does matrix C have a rank of one?

• It is a zero matrix

• It has more columns than rows

• The second row is a multiple of the first row

• It has only one row

What is the relationship between the rows in matrix C?

• One row is a multiple of the other

• They are identical

• They are orthogonal

• They are linearly independent

How many linearly independent columns does matrix D have?

• 1

• 2

• 3

• 4

What is the rank of matrix D?

• 2

• 4

• 3

• 1