# Question about the properties of the rank of a matrix

• skyturnred
In summary, the rank of a matrix is the maximum number of linearly independent rows or columns in the matrix, which can be calculated by counting the number of non-zero rows in the reduced row echelon form of the matrix. It cannot be greater than the number of rows or columns, and it can provide information about the vector space, solutions to linear equations, and invertibility of the matrix. The rank remains unchanged when performing row operations, but may change when performing column operations.
skyturnred

## Homework Statement

Matrix A is a 4 row by 5 column matrix. Matrix B is a column vector in R$^{4}$. We are supposed to decide whether the following are (a) no solution, (b) one solution, or (c) infinitely many solutions, or whether (d) the data do not give enough information to tell, or (e) the data are impossible.

For the following six cases:

rank(A) rank([A | b])
(i) 4 5
(ii) 4 4
(iii) 4 3 .
(iv)3 4
(v) 3 5
(vi)3 3

***numbers on the right are the rank for augmented matrix, numbers on left are for matrix A alone

## The Attempt at a Solution

I THINK I know the answer to all of them.. but I am not entirely certain. I have always found rank just a little confusing. Can someone please tell me where I went wrong in the following and why?

(i) The data is impossible: rank exceeds number of rows.
(ii) There are 5 unkowns (because matrix A has 5 columns) and the rank is 4. So that means there is ONE free variable, so there are infinitely many solutions.
(iii) Not possible: augmenting Matrix A by a column vector B cannot possible DECREASE the rank. Can only be equal to or ONE more than rank(A).
(iv) No solutions. Since augmenting by a column vector increased the rank by one, that means that the augmented matrix is inconsistent.
(v)Not possible: rank exceeds number of rows
(vi)Infinitely many solutions: 5 unknowns, rank is 3, so 2 free variables.

Thanks!

Your answers for (i), (ii), (iv), and (vi) are correct. However, for (iii) and (v), your reasoning is incorrect.

(iii) It is possible for the augmented matrix to have a rank lower than the original matrix. This means that there is a linear dependence between the columns of the original matrix, and adding the column vector B does not change this. Therefore, there are infinitely many solutions.

(v) The rank of the augmented matrix can be equal to the number of rows, as long as the rank of the original matrix is less than the number of rows. In this case, there is a unique solution. However, if the rank of the original matrix and the augmented matrix are both equal to the number of rows, then there are infinitely many solutions. So the correct answer for this case is (c) infinitely many solutions.

## 1. What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it is the number of rows or columns that are needed to create a basis for the vector space spanned by the matrix.

## 2. How is the rank of a matrix calculated?

The rank of a matrix can be calculated by performing row operations on the matrix and counting the number of non-zero rows in the reduced row echelon form of the matrix. The number of non-zero rows is equal to the rank of the matrix.

## 3. Can the rank of a matrix be greater than the number of rows or columns?

The rank of a matrix cannot be greater than the number of rows or columns. This is because the number of linearly independent rows or columns cannot be greater than the total number of rows or columns in the matrix.

## 4. What does the rank of a matrix tell us about its properties?

The rank of a matrix can provide information about the dimension of the vector space spanned by the matrix, the existence of a unique solution to a system of linear equations, and the invertibility of the matrix.

## 5. How does the rank of a matrix change when performing operations on the matrix?

The rank of a matrix remains unchanged when performing row operations such as scaling, swapping, or adding multiples of one row to another. However, it may change when performing column operations as it affects the linear independence of the columns.

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