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Chapter 3: Problem 19

Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be an \(n \times p\) matrix with rank \(n\). Determine the rank of \(A B\). Justify your answer.

Short Answer

Expert verified
The rank of the matrix product AB is equal to the rank of matrix A, which is m. This is because the linearly independent row vectors of A and column vectors of B result in linearly independent row vectors in AB after the matrix multiplication, thus preserving the rank in the product.

Step by step solution

01

Determine the size of the matrix product AB

First, let's determine the size of the matrix product AB. Since matrix A has dimensions m 脳 n and matrix B has dimensions n 脳 p, the matrix product AB will have dimensions m 脳 p.
02

Apply the properties of matrix rank

Now, let's apply the properties of matrix rank to the product AB. We know that rank(AB) 鈮� min(rank(A), rank(B)). Since rank(A) = m and rank(B) = n, we have rank(AB) 鈮� min(m, n). We are trying to determine whether rank(AB) is equal to or less than min(m, n). Let鈥檚 look deeper at matrices A and B: - Matrix A has dimensions m 脳 n and rank m. This means that all row vectors of A are linearly independent. - Matrix B has dimensions n 脳 p and rank n. This means that all column vectors of B are linearly independent.
03

Explain the linear independence of the product

Since all the row vectors of A are linearly independent and all column vectors of B are linearly independent, when we multiply A by B, we are combining the linearly independent vectors in a way that does not introduce any new linear dependence. Thus, the row vectors of the matrix product AB are also linearly independent. Now, since AB has dimensions m 脳 p, and the row vectors of AB are linearly independent, the rank of AB must also be equal to m because the row vectors of AB form a basis of an m-dimensional space.
04

Conclusion and Justification

Therefore, we have determined that the rank of the matrix product AB is equal to the rank of matrix A, which is m. This satisfies the condition rank(AB) 鈮� min(m, n). The reasoning behind our answer is that the linearly independent row vectors of A and column vectors of B result in linearly independent row vectors in AB after the matrix multiplication, thus preserving the rank in the product.

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