If A is 3×2 with two pivot positions, then Ax = b has a solution for every b in R3.
To determine if the equation Ax = b has a solution for every b in R3, we need to analyze the matrix A and its properties
To determine if the equation Ax = b has a solution for every b in R3, we need to analyze the matrix A and its properties.
Given that A is a 3×2 matrix with two pivot positions, we know that A has two leading 1’s in its reduced row echelon form. This means that A has full column rank, as it has exactly two linearly independent columns.
Now, let’s consider the equation Ax = b, where A is a 3×2 matrix and x is a 2×1 column vector. Since the product of A (3×2) and x (2×1) results in a 3×1 column vector, we are trying to solve a system of linear equations with three equations (rows) and one unknown (x).
However, since A has more columns than rows (A is wide), the system is overdetermined, and we expect infinitely many solutions or no solution at all.
To determine if there is a solution for every b in R3, we need to consider the consistency of the system of equations represented by Ax = b. Since A is wide, we can conclude that there are either no solutions or infinitely many solutions.
If A has full column rank, as we determined earlier, then there will be infinitely many solutions for the equation Ax = b. In other words, for every b in R3, there will be a solution to the equation.
Therefore, if A is a 3×2 matrix with two pivot positions, then Ax = b will have a solution for every b in R3.
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