GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 11

Answer 1

A row like $(000 \ldots 0 \mid 0)$ can indicate free variable (and column), but not necessarily infinite solution. We might still find a row of the form: $(00 \cdots 0 0 \mid c).$

Comments

A row like (000…0∣0) can indicate free variable (and column) only when the matrix(not the augmented one ) is square matrix ,cause in square matrix every row must have a pivot element to be a full rank matrix and having zero row in square matrix mean at least 1 free variable present.

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No, a row like (0000...0|0) indicates atleast one free variable. So if solution exits, there would always be infinite solution.

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for unique solution all the column should be independent of matrix A i think.

Answer 2

Cross Example for Statement 1 to prove it False:

This is a Augmented Matrix with 3 variables and 5 equations.

$$\begin{bmatrix} a & b & c & d \\ 0 & e & f & g \\ 0 & 0 & h & i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$
where, all alphabets are non-zero numbers.

We can see that last 2 rows are all zero, but still this equation will have a unique solution
because all the variables are pivot and there are 0 free variables.

For the solution to be infinite, there has to be a free variable.

Statement 2 is True because that is the condition for the No Solution.

For example,
0x = 3
It's a fact that No solution will exist for this.