Determine if Linearly Dependent A={[[2],[4,-5]],[[1,2],[1,10]]}

Problem

A={[[2,0],[4,−5]],[[1,2],[1,10]]}

Solution

1. Set up a linear combination of the matrices and set it equal to the zero matrix.

(c_1)*[[2,0],[4,−5]]+(c_2)*[[1,2],[1,10]]=[[0,0],[0,0]]

2. Create a system of linear equations by comparing the corresponding entries of the matrices.

2*(c_1)+1*(c_2)=0

0*(c_1)+2*(c_2)=0

4*(c_1)+1*(c_2)=0

−5*(c_1)+10*(c_2)=0

3. Solve the system starting with the second equation.

2*(c_2)=0⇒(c_2)=0

4. Substitute the value of (c_2) into the first equation to find (c_1)

2*(c_1)+0=0⇒(c_1)=0

5. Verify if these values satisfy the remaining equations.

4*(0)+0=0

−5*(0)+10*(0)=0

6. Conclude that since the only solution is the trivial solution (c_1)=0 and (c_2)=0 the set is linearly independent.

Final Answer

A=Linearly Independent

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