Question

2. (6 pts) Fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables. ( ) Read Example 5 on page 77 (a)?222-3 2x1-4z2 b) ?? r2-2r3

Answer 1

Solution:

(a) Method I: Let the required matrix be [a d g b c h c f i] Then, [a d g b c h c f i] [x_1 x_2 x_3] = [2x_1 - 4x_2 x_1 + 2x_2 - x_3 x_2 - 2x_3] [a d g b c h c f i] [1 0 0] = [2 1 0] therefore [a d g] = [2 1 0] In the same way [a d g b c h c f i] [0 1 0] = [-4 2 1] therefore [b c h] = [-1 2 1] [a d g b c h c f i] [0 0 1] = [0 -1 -2] therefore [c f i] = [0 -1 -2] therefore The required matrix is [2 1 0 -4 2 1 0 -1 -2] Method II: [2x_1 - 4x_2 x_1 + 2x_2 - x_3 x_2 - 2x_3] = [2 1 0 -4 2 1 0 -1 -2] [x_1 x_2 x_3] (b) Method I: Let the required matrix be

$\small \left ( a \right )$   

Method I:

Let the required matrix be

Then,

$\small \begin{bmatrix} a & b &c \\ d& e &f \\ g & h&i \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3}\\ \end{bmatrix}=\small \begin{bmatrix} 2x_{1}-4x_{2}\\ x_{1}+2x_{2}-x_{3}\\ x_{2}-2x_{3}\\ \end{bmatrix}$

$\small \therefore\begin{bmatrix} a\\ d\\ g\\ \end{bmatrix}=\small \begin{bmatrix} 2\\ 1\\ 0\\ \end{bmatrix}$

In the same way

$\small \begin{bmatrix} a & b &c \\ d& e &f \\ g & h&i \end{bmatrix}\begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix}=\small \begin{bmatrix} -4\\ 2\\ 1\\ \end{bmatrix}$

$\small \therefore \begin{bmatrix} b\\ e\\ h\\ \end{bmatrix}=\small \begin{bmatrix} -4\\ 2\\ 1\\ \end{bmatrix}$

$\small \therefore \begin{bmatrix} c\\ f\\ i\\ \end{bmatrix}=\small \begin{bmatrix} 0\\ -1\\ -2\\ \end{bmatrix}$

$\small \therefore$ The required matrix is

$\small \begin{bmatrix} 2 & -4 &0 \\ 1&2 &-1 \\ 0 &1 & -2 \end{bmatrix}$

Method II:

$\small \begin{bmatrix} 2x_{1}-4x_{2}\\ x_{1}+2x_{2}-x_{3}\\ x_{2}-2x_{3}\\ \end{bmatrix}=\begin{bmatrix} 2 &-4 & 0\\ 1 & 2 &-1 \\ 0 &1 & -2 \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3}\\ \end{bmatrix}$

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$\small \left ( b \right )$

Method I:

Let the required matrix be

$\small \begin{bmatrix} a &b \\ c & d\\ e & f \end{bmatrix}$

Then ,

$\small \begin{bmatrix} a &b \\ c & d\\ e & f \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2}\\ \end{bmatrix}=\begin{bmatrix} 2x_{1}-x_{2}\\ x_{1}+4x_{2}\\ x_{2}\\ \end{bmatrix}$

$\small \begin{bmatrix} a &b \\ c & d\\ e & f \end{bmatrix}\begin{bmatrix} 1\\ 0\\ \end{bmatrix}=\begin{bmatrix} 2\\ 1\\ 0\\ \end{bmatrix}$

$\small \therefore \begin{bmatrix} a\\ c\\ e\\ \end{bmatrix}=\begin{bmatrix} 2\\ 1\\ 0\\ \end{bmatrix}$

$\small \begin{bmatrix} a &b \\ c & d\\ e & f \end{bmatrix}\begin{bmatrix} 0\\ 1\\ \end{bmatrix}=\begin{bmatrix} -1\\ 4\\ 1\\ \end{bmatrix}$

$\small \therefore \begin{bmatrix} b\\ d\\f\\ \end{bmatrix}=\begin{bmatrix} -1\\ 4\\ 1\\ \end{bmatrix}$

$\small \therefore$ The required matrix is

$\small \begin{bmatrix} 2 &-1 \\ 1& 4\\ 0 & 1 \end{bmatrix}$

Method II:

$\small \begin{bmatrix} 2x_{1}-x_{2}\\ x_{1}+4x_{2}\\ x_{2}\\ \end{bmatrix}=\begin{bmatrix} 2 &-1\\ 1 & 4\\ 0& 1\end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2}\\ \end{bmatrix}$