The rank of the matrix ⎛⎝⎜⎜⎜211142346345⎞⎠⎟⎟⎟(246123134145) is

The rank of a matrix is the maximum number of linearly independent rows or columns. To find the rank of a matrix, we can use the method of Gaussian elimination to reduce the matrix to its row echelon form or reduced row echelon form.

The given matrix is:

2 1 1 1 4 2 3 4 6 3 4 5 2 4 6 1 2 3 1 3 4 1 4 5

Step 1: Subtract the first row from the third row:

2 1 1 1 4 2 3 4 6 3 4 5 0 3 5 0 -2 1 1 3 4 1 4 5

Step 2: Subtract the first row from the fourth row:

2 1 1 1 4 2 3 4 6 3 4 5 0 3 5 0 -2 1 -1 2 3 0 0 3

Step 3: Subtract the second row from the third row:

2 1 1 1 4 2 3 4 6 3 4 5 -3 -1 -1 -3 -6 -4 -1 2 3 0 0 3

Step 4: Subtract the second row from the fourth row:

2 1 1 1 4 2 3 4 6 3 4 5 -3 -1 -1 -3 -6 -4 -4 -2 -3 -3 -4 -2

Step 5: Multiply the third row by -1:

2 1 1 1 4 2 3 4 6 3 4 5 3 1 1 3 6 4 -4 -2 -3 -3 -4 -2

Step 6: Add the third row to the fourth row:

2 1 1 1 4 2 3 4 6 3 4 5 3 1 1 3 6 4 -1 -1 -2 0 2 2

Step 7: Multiply the fourth row by -1:

2 1 1 1 4 2 3 4 6 3 4 5 3 1 1 3 6 4 1 1 2 0 -2 -2

Now, we can see that the first, second, and third rows are linearly independent, but the fourth row is a linear combination of the first and second rows. Therefore, the rank of the matrix is 3.