Determine if Linearly Dependent A={[[1,2],[3,4]],[[1,2],[12,2]]}

Problem

A={[[1,2],[3,4]],[[1,2],[12,2]]}

Solution

1. Define the condition for linear dependence. Two matrices (M_1) and (M_2) are linearly dependent if there exist scalars (c_1) and (c_2) not both zero, such that (c_1)*(M_1)+(c_2)*(M_2)=0 For a set of two vectors, this is equivalent to checking if one matrix is a scalar multiple of the other.

2. Set up the scalar equation. We check if there exists a scalar k such that:

[[1,2],[12,2]]=k*[[1,2],[3,4]]

3. Compare corresponding entries to find k From the entry in the first row, first column:

1=k(1)⇒k=1

4. Verify the scalar for all other entries. Check the entry in the second row, first column using k=1

12=1*(3)

12=3

5. Conclude based on the contradiction. Since 12≠3 there is no single scalar k that satisfies the equation for all entries. Therefore, the matrices are not scalar multiples of each other.

Final Answer

The set *A* is Linearly Independent.

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