Elementary Transformation of a matrix:
The following operation on a matrix are called elementary operations (transformations)
1. The interchange of any two rows (or columns)
2. The multiplication of the elements of any row (or column) by any nonzero number
3. The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number
Echelon Form of matrix : A matrix A is said to be in echelon form if
(i) every row of A which has all its elements 0, occurs below row, which has a non-zero elements
(ii) the first non-zero element in each non –zero row is 1.
(iii) The number of zeros before the first non zero elements in a row is less than the number of such zeros in the next now.
[ A row of a matrix is said to be a zero row if all its elements are zero]
Note: Rank of a matrix does not change by application of any elementary operations
For example `[(1,1,3),(0,1,2),(0,0,0)],[(1,1,3,6),(0,1,2,2),(0,0,0,0)]` are echelon forms
The number of non-zero rows in the echelon form of a matrix is defined as its RANK. For example we can reduce the matrix `A=[(1,2,3),(2,4,7),(3,6,10)]` into echelon form using following elementary row transformation.
(i)`R_2 to R_2 -2R_1` and `R_3 to R_3 -3R_1 [(1,2,3),(0,0,1),(0,0,1)]`
(ii)`R_2 to R_2 -2R_1 [(1,2,3),(0,0,1),(0,0,0)]`
This is the echelon form of matrix A Number of nonzero rows in the echelon form =2 `rArr` Rank of the matrix A is 2
Rank of the matrix `[(1,1,1,-1),(1,2,4,4),(3,4,5,2)]` is :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the rank of the matrix \( A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 1 & 2 & 4 & 4 \\ 3 & 4 & 5 & 2 \end{pmatrix} \), we will perform elementary row transformations to convert it into echelon form.

### Step 1: Apply the first row operation
We will perform the operation \( R_2 \leftarrow R_2 - R_1 \) and \( R_3 \leftarrow R_3 - 3R_1 \).

- For \( R_2 \):
\[ R_2 = (1, 2, 4, 4) - (1, 1, 1, -1) = (1-1, 2-1, 4-1, 4-(-1)) = (0, 1, 3, 5) \]

- For \( R_3 \):
\[ R_3 = (3, 4, 5, 2) - 3 \times (1, 1, 1, -1) = (3-3, 4-3, 5-3, 2-(-3)) = (0, 1, 2, 5) \]

Now the matrix looks like:
\[ A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 1 & 3 & 5 \\ 0 & 1 & 2 & 5 \end{pmatrix} \]

### Step 2: Apply the second row operation
Next, we will perform the operation \( R_3 \leftarrow R_3 - R_2 \).

- For \( R_3 \):
\[ R_3 = (0, 1, 2, 5) - (0, 1, 3, 5) = (0-0, 1-1, 2-3, 5-5) = (0, 0, -1, 0) \]

Now the matrix looks like:
\[ A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & -1 & 0 \end{pmatrix} \]

### Step 3: Simplify the third row
To make the leading coefficient of the third row a positive 1, we can multiply \( R_3 \) by -1:
\[ R_3 \leftarrow -R_3 = (0, 0, 1, 0) \]

Now the matrix looks like:
\[ A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 1 & 0 \end{pmatrix} \]

### Step 4: Count the non-zero rows
Now we can see that the echelon form of the matrix has 3 non-zero rows:
1. \( (1, 1, 1, -1) \)
2. \( (0, 1, 3, 5) \)
3. \( (0, 0, 1, 0) \)

### Conclusion
The rank of the matrix \( A \) is the number of non-zero rows in its echelon form, which is 3.

Thus, the rank of the matrix \( A \) is **3**.

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