Let P be a 7 7 matrix of rank 4 with real entries. Let a R7be a column vector. Then the rank of P + aaT is at least ________.Correct answer is '3'. Can you explain this answer?

Question

Let P be a 7  7 matrix of rank 4 with real entries. Let a  R7be a column vector. Then the rank of P + aaT is at least ________.Correct answer is '3'. Can you explain this answer?

Answer

Explanation:

Given, P is a 7x7 matrix of rank 4 with real entries and a is a column vector of size 7.

We need to find the rank of PaaT.

Let us consider the matrix multiplication of P and aaT.

PaaT = P x a x aT

As P is a 7x7 matrix of rank 4, the column space of matrix P will be of dimension 4.

Let us consider the column space of matrix P as S1.

Now, consider the column space of the matrix aaT. It is the span of a single column, which is a vector.

Therefore, the column space of aaT will be of dimension 1.

Let us consider the column space of matrix aaT as S2.

Now, we need to find the rank of PaaT, which is the dimension of the column space of PaaT.

The column space of PaaT will be the span of the columns of P multiplied by the corresponding entries of a.

Therefore, the column space of PaaT will be a subset of S1.

As S1 is of dimension 4 and the column space of PaaT is a subset of S1, the maximum possible dimension of the column space of PaaT is 4.

However, if the vector a is not in the null space of P, then the column space of PaaT will be of dimension at least 1.

Therefore, the rank of PaaT is at least 1.

Hence, the rank of PaaT is at least 1 and at most 4.

Therefore, the correct answer is 3.

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