Find the Rank [[a-b,b+c],[-3d+c,2a-4d]]

Problem

rank(a−b)

Solution

1. Represent the matrix A as a linear combination of basis matrices by separating the variables a,b,c, and d

A=a*[[1,0],[0,2]]+b*[[−1,1],[0,0]]+c*[[0,1],[1,0]]+d*[[0,0],[−3,−4]]

2. Vectorize the basis matrices into vectors in ℝ4 to form a new matrix M where each column represents one variable's contribution.

M=[[1,−1,0,0],[0,1,1,0],[0,0,1,−3],[2,0,0,−4]]

3. Apply row reduction to M to find its rank, which corresponds to the maximum possible rank of the original matrix transformation.

(R_4)→(R_4)−2*(R_1)⇒[[1,−1,0,0],[0,1,1,0],[0,0,1,−3],[0,2,0,−4]]

4. Continue row reduction to reach row echelon form.

(R_4)→(R_4)−2*(R_2)⇒[[1,−1,0,0],[0,1,1,0],[0,0,1,−3],[0,0,−2,−4]]

(R_4)→(R_4)+2*(R_3)⇒[[1,−1,0,0],[0,1,1,0],[0,0,1,−3],[0,0,0,−10]]

5. Determine the rank by counting the number of non-zero rows. Since there are 4 linearly independent basis matrices, the dimension of the subspace is 4. However, the rank of a 2×2 matrix is at most 2.

6. Evaluate the rank of the specific 2×2 matrix. The rank of a matrix is the number of linearly independent rows or columns. For a 2×2 matrix, the rank is 2 if the determinant is non-zero, 1 if the matrix is non-zero but the determinant is zero, and 0 if all entries are zero.

Final Answer

rank(a−b)≤2

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