Find the Rank [[1,h,4],[-5,10,-24]]

Problem

Rank*[[1,h,4],[−5,10,−24]]

Solution

1. Identify the dimensions of the matrix. The matrix has 2 rows and 3 columns, so its maximum possible rank is 2.

2. Apply row operations to transform the matrix into row-echelon form. To eliminate the first element of the second row, add 5 times the first row to the second row ((R_2)→(R_2)+5*(R_1).

3. Calculate the new second row. The first element becomes −5+5*(1)=0 The second element becomes $10 + 5(h) = 10 + 5h.T*h*e*t*h*i*r*d(e)*l*e*m*e*n*t*b*e*c*o*m*e*s()24 + 5(4) = -4$.

4. Analyze the resulting matrix. The matrix is now:

[[1,h,4],[0,10+5*h,−4]]

5. Determine the rank by counting the number of non-zero rows. Since the first row has a leading 1 and the second row has a non-zero constant −4 in the third column, the second row can never be a row of all zeros, regardless of the value of h

6. Conclude that there are always 2 linearly independent rows.

Final Answer

Rank*[[1,h,4],[−5,10,−24]]=2

Want more problems? Check here!