Please show your work Given the following matrices A and B f

Please show your work.

Given the following matrices A and B, find a matrix C in M_2, 3 so that {A, B, C} is linearly independent and a non-zero matrix Din M_2, 3 so that {A, B, D} is linearly dependent: A = [-4 8 9 3 9 -9] B = [10 7 4 0 7 -8] C = [0 0 0 0 0 0] D = [0 0 0 0 0 0]

Solution

To find C such that A,B,C is linearly independent we know that

A(x) + B(y) + C(z) = 0

A(x) and B(y) are known so

The set S is not linearly dependent if, whenever some linear combination
of the elements of S adds up to 0, it turns out that c₁, c₂, . . . are all zero. That is,
c₁x₁ + · · · + c_nx_n = 0

c₁ = c₂ = · · · = c_n = 0

Suppose that, in the matrix A, row_i(A) is replaced by row_i(A) + c·row_j (A). Call the resulting
matrix B. If x belongs to the row space of A, then
x = c₁row₁(A) + . . . + c_irow_i(A) + . . . + c_j row_j (A) + c_mrow_m(A).
Now add and subtract c · c_i· row_j (A) to get
x = c₁row₁(A) + . . . + c_irow_i(A) + c · c_irow_j (A) + . . . + (c_j c_i· c)row_j (A) + c_mrow_m(A)
= c₁row₁(B) + . . . + c_irow_i(B) + . . . + (c_j c_i· c)row_j (B) + . . . + c_mrow_m(B).
This shows that x can also be written as a linear combination of the rows of B. So any
element in the row space of A is contained in the row space of B