Understanding the Solvability of the Equation Ax = b in R3: Importance of Linear Independence and Matrix Rank

If A is 3×3 with three pivot positions, then Ax = b has a solution for every b in R3.

To determine whether the equation Ax = b has a solution for every b in R3, where A is a 3×3 matrix with three pivot positions, we need to consider the concept of linear independence and the rank of the matrix

To determine whether the equation Ax = b has a solution for every b in R3, where A is a 3×3 matrix with three pivot positions, we need to consider the concept of linear independence and the rank of the matrix.

A pivot position in a matrix represents a nonzero entry in a row, which leads to linearly independent rows. If the matrix A has three pivot positions, it means that all three rows of A are linearly independent.

When the matrix A has linearly independent rows, it is said to have full rank. In this case, A has a rank of 3, which is the maximum possible rank for a 3×3 matrix.

Now, let’s consider the equation Ax = b.

If A has a full rank of 3, it means that each R3 vector can be expressed as a linear combination of the columns of A. Therefore, for any given b in R3, we can find a unique solution x to the equation Ax = b. This shows that the equation Ax = b has a solution for every b in R3.

In conclusion, if A is a 3×3 matrix with three pivot positions, then the equation Ax = b has a solution for every b in R3.

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