Math

Question Find the rank of 2×22 \times 2 matrix AA if Ax=0Ax=0 has only the zero solution.

Studdy Solution

STEP 1

Assumptions
1. AA is a 2×22 \times 2 matrix.
2. AA is nonzero.
3. The equation Ax=0A x = 0 has only the zero solution, where xx is a column vector.

STEP 2

The fact that Ax=0A x = 0 has only the zero solution implies that the null space of AA contains only the zero vector.

STEP 3

The null space of a matrix is the set of all vectors that are mapped to the zero vector by the matrix.

STEP 4

If the null space contains only the zero vector, then the matrix is of full rank, meaning that its columns are linearly independent.

STEP 5

For a 2×22 \times 2 matrix to be of full rank, its rank must be equal to the number of columns, which is 2.

STEP 6

Therefore, we can conclude that the rank of matrix AA is 2.
rank(A)=2\operatorname{rank}(A) = 2

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