Here is the question I want to prove:
Let $V = M_2(\mathbb{R})$ be the space of all $2\times 2$ matrices over $\mathbb{R}.$ Show that $$B(X,Y) = det(X+Y) - det(X) - det(Y)$$
Where $X,Y \in V,$ is a symmetric nondegenerate bilinear form on $V.$
I tried to prove linearity but it did not work. Also, I was unable to prove nondegeneracy. Could someone tell me if there is a typo in this question please?
There's no typo in the question. If $$ X=\pmatrix{x_{11}&x_{12}\\x_{21}&x_{22}}\ ,\ Y=\pmatrix{y_{11}&y_{12}\\y_{21}&y_{22}}\ , $$ then \begin{align} B(X,Y)&=\big(x_{11}+y_{11}\big)\big(x_{22}+y_{22}\big)-\big(x_{12}+y_{12}\big)\big(x_{21}+y_{21}\big)\\ &\hspace{3em}-x_{11}x_{22}+x_{12}x_{21}-y_{11}y_{22}+y_{12}y_{21}\\ &=x_{11}y_{22}+x_{22}y_{11}-x_{12}y_{21}-x_{21}y_{12}\\ &=\pmatrix{x_{11}&x_{12}&x_{21}&x_{22}}\pmatrix{0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0}\pmatrix{y_{11}\\y_{12}\\y_{21}\\y_{22}}\ . \end{align} It follows that $\ B\ $ is bilinear, and $$ M=\pmatrix{0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0} $$ is its matrix with respect to the basis $$ \left\{\pmatrix{1&0\\0&0}, \pmatrix{0&1\\0&0},\pmatrix{0&0\\1&0},\pmatrix{0&0\\0&1}\right\} $$ of $\ M_2(\mathbb{R})\ $. The symmetry and non-degeneracy of $\ B\ $ follow from those of its matrix $\ M\ $.