row operations

**Lane** · July 22nd 2006, 07:05 AM

questions on attachment!!

**Soroban** · July 22nd 2006, 08:26 AM

Hello, Lane!

Here's some help . . .

1) What is the augmented matrix for: . $\begin{array}{ccc}9x + 3y - 9z\,=\,\text{-}2 \\ 3x + 7y - 7z\,=\,6 \\ \text{-}9x - 8y - 5x\,=\,\text{-}4\end{array}$

. . $\begin{pmatrix}9 & 3 & \text{-}9 & | & \text{-}2 \\ 3 & 7 & \text{-}7 & | & 6 \\ \text{-}9 & \text{-}8 & \text{-}5 & | & \text{-}4\end{pmatrix}$

2) Perform the row operation on the matrix: . $\begin{pmatrix}\text{-}8 & 1 & \text{-}4 & | & 6 \\ \text{-}7 & \text{-}5 & \text{-}3 & | & 4 \\ 0 & 7 & \text{-}1 & | & 3\end{pmatrix}$
. . Multiply $R_2$ by $4.$

. . $\begin{array}{cccc} \\ 4\!\cdot\!R_2 \\ \\ \end{array}\,\begin{pmatrix}\text{-}8 & 1 & \text{-}4 & | & 6 \\ \text{-}28 & \text{-}20 & \text{-}12 & | & 16 \\ 0 & 7 & \text{-}1 & | & 3\end{pmatrix}$

3) Perform the sequence of row operations on: . $\begin{pmatrix}3 & 10 & \text{-}1 \\ 1 & 5 & \text{-}9 \\ \text{-}4 & \text{-}7 & 1\end{pmatrix}$
A: swap $R_1$ and $R_2$
B: replace $R_2$ by $R_2$ plus $\text{-}3$ times $R_1$
C: replace $R_3$ by $R_3$ plus $4$ times $R_1$

. . $\begin{array}{cccc}R_1\leftrightarrow R_2\\ \\ \\ \end{array}\,\begin{pmatrix}1 & 5 & \text{-}9 \\ 3 & 10 & \text{-}1 \\ \text{-}4 & \text{-}7 & 1\end{pmatrix}$

. . $\begin{array}{cccc} \\ R_2- 3\!\cdot\!R_1 \\ \\ \end{array}\,\begin{pmatrix} 1 & 5 & \text{-}9 \\ 0 & \text{-}5 & 26 \\ \text{-}4 & \text{-}7 & 1\end{pmatrix}$

. . $\begin{array}{ccc} \\ \\ R_3 + 4\!\cdot\!R_1\end{array}\,\begin{pmatrix}1 & 5 & \text{-}9 \\ 0 & \text{-}5 & 26 \\ 0 & 13 & \text{-}35\end{pmatrix}$

4) If possible, solve the system of equation using Gaussian elimination with back-substitution.

. . $\begin{array}{ccc}x + 3y + z\,=\,-12 \\ 2x + 7y - 2z \,= \,-25 \\ x - y - 6z\,=\,15\end{array}$

The determinant is $\text{-}23$ . . . I expect a very messy elimination.
I'll let someone else work on it . . .

5) If possible solve the system of equation using an inverse matrix.

. . $\begin{array}{ccc}3x + 2y - 2x\,=\,-8 \\ 3x - y - 3x\,=\,-4 \\ 6x + y - 5z\,=\,6\end{array}$

Not possible . . . the determinant is $0$ .

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