Question

For each of the following matrices, use Gaussian elimination on the augmented matrix $(A \mid \mathbf{b})$ to determine a basis for its cokernel: (a) $\left(\begin{array}{ll}9 & -6 \\ 6 & -4\end{array}\right)$, (b) $\left(\begin{array}{rr}1 & 3 \\ 2 & 6 \\ -3 & -9\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & 1 & 3 \\ -1 & 1 & -2 \\ -1 & 3 & 6\end{array}\right)$, (d) $\left(\begin{array}{rrr}1 & -2 & -2 \\ 0 & -1 & 3 \\ 2 & -5 & -1 \\ -2 & 2 & 10\end{array}\right)$.

   For each of the following matrices, use Gaussian elimination on the augmented matrix $(A \mid \mathbf{b})$ to determine a basis for its cokernel:
(a) $\left(\begin{array}{ll}9 & -6 \\ 6 & -4\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & 3 \\ 2 & 6 \\ -3 & -9\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 1 & 3 \\ -1 & 1 & -2 \\ -1 & 3 & 6\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & -2 & -2 \\ 0 & -1 & 3 \\ 2 & -5 & -1 \\ -2 & 2 & 10\end{array}\right)$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 4, Problem 30

Instant Answer

Step 1

Here, the second column does not have a pivot, so $x_2$ is free. **Step 4:** Write the solution in terms of the free variables: - $3x_1 - 2x_2 = 0 \Rightarrow x_1 = \frac{2}{3}x_2$ **Step 5:** Express the basis for the cokernel: - Set $x_2 = t$, then $x_1 = Show more…

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For each of the following matrices, use Gaussian elimination on the augmented matrix $(A \mid \mathbf{b})$ to determine a basis for its cokernel: (a) $\left(\begin{array}{ll}9 & -6 \\ 6 & -4\end{array}\right)$, (b) $\left(\begin{array}{rr}1 & 3 \\ 2 & 6 \\ -3 & -9\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & 1 & 3 \\ -1 & 1 & -2 \\ -1 & 3 & 6\end{array}\right)$, (d) $\left(\begin{array}{rrr}1 & -2 & -2 \\ 0 & -1 & 3 \\ 2 & -5 & -1 \\ -2 & 2 & 10\end{array}\right)$.

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