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I don't know how to do this question, I appreciate if anyone can help. Thank you.

(a) For what values of $a,b$, and $c$ is the following matrix symmetric? $$\begin{bmatrix} -3 & 6a-c & 6a+2b\\ a & 2 & 4\\ a+7b & c & a \end{bmatrix}$$ (b) An $n\times n$ matrix $A$ is called skew-symmetric if $A^T = -A$. What values of $a,b, c,$ and $d$ now make the following matrix skew-symmetric? $$\begin{bmatrix} d & 6a-c & 6a+2b\\ a & 0 & 4-5d\\ a+7b &c &0 \end{bmatrix}$$

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    Please use [MathJax](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference). – 2017-02-07
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    @ Max I would use mathJax but I'm not familiar with coding so forgive me about this issue. Thank you – 2017-02-07

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A symmetric matrix $A$ means that all entries $A_{ij}$ are symmetric respect to the diagonal, i.e., $A_{ij}=A_{ji}$ for all $i$, $j$.

In your case, you need $6a-c=a$, $6a+2b=a+7b$, and $c=4$ for the first matrix to be symmetric. From this, you can solve for $a,b,c$.

In the second case, the definition of skew-symmetric can be written as that $A_{ij}=-A_{ji}$, so you can similarly equate each term to its equivalent in the transposed neagtive matrix.

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    Thanks for your help. Can you please tell me more regarding part B? do I have C=-(4-5d)? then what would be value of C? I got little bit confuse – 2017-02-07
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    Yes, $c=-(4-5d)$, and also $a=-(6a-c)$ and so on. With this, you'll set up a system of four equations and for unknowns that you should be able to solve easily. – 2017-02-07
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    What is fourth equation? d is -1? – 2017-02-07
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    No, if you compute $A^T$ you get $d$ as the first element, and $-A$ has $-d$ as first element, so $-d=d$. – 2017-02-07
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    so when you say -d = d.- means d = d? what that negative sign do after d? ( sorry if I ask basic or funny questions) – 2017-02-07
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    No, it means that $d$ is a number such that $-d=d$, so $d$ is equal to its own negative. Which number can it be? – 2017-02-07
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    Anna@ thing is we have 3 questions. a=-6a+c, c=-4+5d, a+7b=-6a-2b. and the fourth equation should be -d=d. am I right? if I'm right what would be value of d? can you please show me how you would do the second part? – 2017-02-08
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    How many numbers do you know that are the negative of themselves? Let's see: is -1 equal to 1? is -2 equal to 2? is -1/3 equal to 1/3? Is any $d>0$ equal to $-d$? And any $d<0$? – 2017-02-08