Matrix Dimensions in Ax=0 | Generated by AI
Not necessarily! Let me clarify:
x is a column vector with n components: x₁, x₂, …, xₙ. That part is correct.
However, A doesn’t have to be an n×n matrix. A can be an m×n matrix, where:
- m = number of rows (number of equations)
- n = number of columns (number of variables)
So the system Ax = 0 has:
- m equations
- n unknowns (the components of x)
Examples:
- If A is 3×3 and x is 3×1, you have 3 equations in 3 unknowns.
- If A is 5×3 and x is 3×1, you have 5 equations in 3 unknowns (overdetermined system).
- If A is 2×4 and x is 4×1, you have 2 equations in 4 unknowns (underdetermined system).
Special case: If A is specifically an n×n square matrix, then yes, you have n equations in n unknowns, and the behavior of solutions depends on whether A is invertible or singular.
So the general answer is: A is an m×n matrix, and x is an n×1 column vector, but they don’t have to be square!