Let P and Q be two real matrices of size 4 6 and 5 4, respectively. If rank(Q) = 4 and rank(QP) = 2, then rank(P) is equal to ______Correct answer is '2'. Can you explain this answer?

Question

Let P and Q be two real matrices of size 4  6 and 5  4, respectively. If rank(Q) = 4 and rank(QP) = 2, then rank(P) is equal to ______Correct answer is '2'. Can you explain this answer?

Answer

Explanation:

Rank of a matrix is the maximum number of linearly independent rows or columns in a matrix.

Given,
- P is a matrix of size 4 x 6
- Q is a matrix of size 5 x 4
- rank(Q) = 4
- rank(QP) = 2

We need to find rank(P).

Step 1: Finding the nullity of Q

Nullity of Q = number of columns in Q - rank(Q)

Nullity of Q = 4 - 4 = 0

Step 2: Using rank-nullity theorem to find the rank of P

rank(QP) = rank(P) + nullity(Q)

2 = rank(P) + 0

rank(P) = 2

Therefore, the rank of P is 2.

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