Find the Rank (2A^T)^-1=[[4,5],[3,4]]

Problem

(2*AT)(−1)=[[4,5],[3,4]]

Solution

1. Identify the given matrix M=(2*AT)(−1)

2. Recall the property that the rank of a matrix is equal to the rank of its inverse, provided the inverse exists.

3. Recall the property that the rank of a matrix k*M (where k≠0 is equal to the rank of M

4. Recall the property that the rank of a matrix A is equal to the rank of its transpose AT

5. Calculate the determinant of the given matrix to determine if it is non-singular.

det((2*AT)(−1))=(4)*(4)−(5)*(3)

det((2*AT)(−1))=16−15

det((2*AT)(−1))=1

6. Conclude that since the determinant is non-zero, the 2×2 matrix is of full rank.

rank((2*AT)(−1))=2

7. Relate the rank of the inverse back to the original matrix A Since (2*AT)(−1) has rank 2, then 2*AT must have rank 2, and consequently A must have rank 2.

Final Answer

rank(A)=2

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