Find the conditions required for the values of a, b, and c that make the following matrix symmetric.

Question

Set up the system:

$$A = \begin{bmatrix} 5& a+b+c& a-b \\ 3& -7& 2\\ 1& a+c & 6 \end{bmatrix}$$

I did it like this: \begin{bmatrix} 5& 3& 1 \\ 3& -7& a+c\\ 1& a+c & 6 \end{bmatrix}

Did I answer the question correctly? Any help is very much appreciated!

Answer 1

You have not solved for $a$, $b$ and $c$. As you have right observed, we need $A_{ij} = A_{ji}$ for the matrix to be symmetric. Hence, we need $$a+b+c = 3$$ $$a-b = 1$$ $$a+c = 2$$ You have three equations in three unknowns. Solve for $a$, $b$ and $c$.

Answer 2

Solve a linear system with euqations

a + b + c = 3

a - b = 1

a + c = 2

Answer 3

To find the conditions you must have

\begin{bmatrix} 5& a+b+c& a-b \\ 3& -7& 2\\ 1& a+c & 6 \end{bmatrix}=\begin{bmatrix} 5& 3& 1 \\ 3& -7& a+c\\ 1& a+c & 6 \end{bmatrix}.

This gives $a+b+c=3$, $a-b=1$ and $a+c=2$. This would give $a=2$,$b=1$ and $c=0$.