# solving a matrix equation for a,b,c, and d

• Feb 3rd 2010, 10:52 AM
cottekr
solving a matrix equation for a,b,c, and d
[a-b b+c] = [8 1]
[3d+c 2a-4d] [7 6]

I know that since the matrices are equal:

a-b=8
b+c=1
3d+c=7
2a-4d=6

but not sure where to go from here
• Feb 3rd 2010, 12:16 PM
pickslides
Quote:

Originally Posted by cottekr
[a-b b+c] = [8 1]
[3d+c 2a-4d] [7 6]

I know that since the matrices are equal:

a-b=8
b+c=1
3d+c=7
2a-4d=6

but not sure where to go from here

You have made a good start.

From the first equation $a = 8+b$

From the second $c = 1-b$

From the third (and second) $d = \frac{7-c}{3} \implies \frac{7-(1-b)}{3} = \frac{6+b}{3}$

Now looking at the 4th equation $2a-4d=6$ we have new information for $a$ and $d$ so lets use it in the 4th.

$
2a-4d=6\implies 2(8+b)-4\left(\frac{6+b}{3}\right)=6
$

Now expand and solve for $b$
• Feb 3rd 2010, 08:08 PM
math2009
$
A\begin{bmatrix}a\\b\\c\\d\end{bmatrix}= \begin{bmatrix}a-b\\b+c\\3d+c\\2a-4d\end{bmatrix}=
a\begin{bmatrix}1\\0\\0\\2\end{bmatrix}+
b\begin{bmatrix}-1\\1\\0\\0\end{bmatrix}+
c\begin{bmatrix}0\\1\\1\\0\end{bmatrix}+
d\begin{bmatrix}0\\0\\3\\-4\end{bmatrix} \\
$

$
=\begin{bmatrix}
1 &-1 &0 &0 \\
0 &1 &1 &0 \\
0 &0 &1 &3 \\
2 &0 &0 &-4 \\
\end{bmatrix}
\begin{bmatrix}a\\b\\c\\d\end{bmatrix}=
\begin{bmatrix}8\\1\\7\\6\end{bmatrix}~,~
\begin{bmatrix}a\\b\\c\\d\end{bmatrix}=
A^{-1}\begin{bmatrix}8\\1\\7\\6\end{bmatrix}=
\begin{bmatrix}5\\-3\\4\\1\end{bmatrix}

$