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Math  /  Algebra

QuestionIf the null space of a 7×57 \times 5 matrix is 4 -dimensional, find rank AA, dim⁡Row⁡A\operatorname{dim} \operatorname{Row} A, and dim⁡Col⁡A\operatorname{dim} \operatorname{Col} A. A. rank A=1,dim⁡Row⁡A=4,dim⁡Col⁡A=4A=1, \operatorname{dim} \operatorname{Row} A=4, \operatorname{dim} \operatorname{Col} A=4 B. rank A=1,dim⁡Row⁡A=1,dim⁡Col⁡A=4A=1, \operatorname{dim} \operatorname{Row} A=1, \operatorname{dim} \operatorname{Col} A=4 C. rank A=3,dim⁡Row⁡A=3,dim⁡Col⁡A=3A=3, \operatorname{dim} \operatorname{Row} A=3, \operatorname{dim} \operatorname{Col} A=3 D. rank⁡A=1,dim⁡Row⁡A=1,dim⁡Col⁡A=1\operatorname{rank} A=1, \operatorname{dim} \operatorname{Row} A=1, \operatorname{dim} \operatorname{Col} A=1

Studdy Solution

STEP 1

1. The matrix A A is a 7×5 7 \times 5 matrix.
2. The null space of A A is 4-dimensional.
3. The rank-nullity theorem applies, which states that for a matrix A A with n n columns, rank(A)+nullity(A)=n\text{rank}(A) + \text{nullity}(A) = n.

STEP 2

1. Use the rank-nullity theorem to find the rank of A A .
2. Determine dim⁡Row⁡A\operatorname{dim} \operatorname{Row} A.
3. Determine dim⁡Col⁡A\operatorname{dim} \operatorname{Col} A.
4. Compare results with the given options.

STEP 3

Apply the rank-nullity theorem. For a 7×5 7 \times 5 matrix A A , the number of columns n=5 n = 5 . The nullity of A A is given as 4. Therefore, we can write:
rank(A)+nullity(A)=n\text{rank}(A) + \text{nullity}(A) = n rank(A)+4=5\text{rank}(A) + 4 = 5
Solve for rank(A)\text{rank}(A):
rank(A)=5−4=1\text{rank}(A) = 5 - 4 = 1

STEP 4

The dimension of the row space of A A is equal to the rank of A A . Therefore:
dim⁡Row⁡A=rank(A)=1\operatorname{dim} \operatorname{Row} A = \text{rank}(A) = 1

STEP 5

The dimension of the column space of A A is also equal to the rank of A A . Therefore:
dim⁡Col⁡A=rank(A)=1\operatorname{dim} \operatorname{Col} A = \text{rank}(A) = 1

STEP 6

Compare the results with the given options:
- rank A=1 A = 1 - dim⁡Row⁡A=1\operatorname{dim} \operatorname{Row} A = 1 - dim⁡Col⁡A=1\operatorname{dim} \operatorname{Col} A = 1
The correct option is:
D. rank⁡A=1,dim⁡Row⁡A=1,dim⁡Col⁡A=1\operatorname{rank} A=1, \operatorname{dim} \operatorname{Row} A=1, \operatorname{dim} \operatorname{Col} A=1
The correct answer is:
D \boxed{D}

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