Suppose matrix A is m Times n and B is n Times p If the firs

Suppose matrix A is m Times n and B is n Times p. If the first row of matrix A is all zeros, what can you say about the first row of AB? Explain. Suppose matrix A is m Times n and B is n Times p. Prove that if the columns of B are linearly dependent, then the columns of AB are also linearly dependent.

Solution

1.
1st element of 1st row of AB is formed by multiplying 1st row of A with corresponding elements of 1st column of B . It will be 0

2nd element of 1st row of AB is formed by multiplying 1st row of A with corresponding elements of 2nd column of B . It will be 0

3rd element of 1st row of AB is formed by multiplying 1st row of A with corresponding elements of 3rd column of B . It will be 0

So 1st row of AB is 0

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