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Suppose $A$ is a $4 \times 3$ matrix and $\mathbf{b}$ is a vector in $\mathbb{R}^{4}$ with the property that $A \mathbf{x}=\mathbf{b}$ has a unique solution. What can you say about the reduced echelon form of $A ?$ Justify your answer.

$\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1} \\ {0} & {0} & {0}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

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University of Michigan - Ann Arbor

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in this video, we start out with the four by three matrix, and we're going to consider the equation. X equals B. Let's start off by making an assumption. Let's suppose that the Matrix Equation A X equals B has a unique solution for some vector be in our for So let's next. Consider what that would specifically mean for this matrix. Well, it would imply that in the equation, X equals B If we have a unique solution, there are no free variables, so start there. So that's last line implies that there are no free variables. How do we know this? Well, if there was a single free variable, we would have infinitely many solutions, not a unique one. So next, knowing that there are no free variables that implies the following. Ah, free variable will always come from a non pivot column of The Matrix, say, so we know that A has a pivot in all Collins, and there's three com specifically. So let's say all three columns squeezing the three in there. Sophie has a pivot in all three columns, but it is of size four by three. This is our next immediate conclusion. I will say that a must be equal to or rather not equal but row equivalents to a four by three matrix. That would look like this so we can have 100 now. Calm One is a pivot column for row to go 010 Now we have to pivot columns, but we require three. So the third row will be 001 and we're looking at the reduced row echelon form, so their final row will be all zeros. So given just this minimal set of information, we now know that the role reduced echelon form of a takes on this form.

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