Question: Show, by multiplying the matrices, that the following equation represents an ellipse.

$\mathbf{\left(}\begin{array}{cc}\mathbf{x}& \mathbf{y}\end{array}\mathbf{\right)}\mathbf{\left(}\begin{array}{cc}\mathbf{5}& \mathbf{-}\mathbf{7}\\ \mathbf{7}& \mathbf{3}\end{array}\mathbf{\right)}\mathbf{\left(}\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\mathbf{\right)}\mathbf{=}\mathbf{30}$

Matrix multiplication is abinary operation that creates a matrix by multiplying two matrices together. The number of columns in the first matrix must equal the number of rows in the second matrix for matrix multiplication to work.

The given equation is $\left(\begin{array}{cc}\mathrm{x}& \mathrm{y}\end{array}\right)\left(\begin{array}{cc}5& -7\\ 7& 3\end{array}\right)\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=30$.

Prove that the above equation represents an ellipse.

Find the product of $\left(\begin{array}{cc}5& -7\\ 7& 3\end{array}\right)\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$.

role="math" localid="1664178381820" $\left(\begin{array}{cc}5& -7\\ 7& 3\end{array}\right)\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=\left(\begin{array}{cc}5x& -7x\\ 7y& +3y\end{array}\right)$

Find the product of $\left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}5x& -7x\\ 7y& +3y\end{array}\right)$.

$\left(\begin{array}{cc}x& y\end{array}\right)\left(\begin{array}{cc}5x& -7x\\ 7y& +3y\end{array}\right)=x(5x-7y)+y(7x+3y)$

Simplify the given equation further.

$\begin{array}{rcl}x(5x-7y)+y(7x+3y)& =& 30\\ 5x^2-7xy+7xy+3y^2& =& 30\\ 5x^2+3y^2& =& 30\end{array}$

Divide by 30 to both sides.

$\begin{array}{rcl}\frac{5x^2}{30}+\frac{3y^2}{30}& =& \frac{30}{30}\\ \frac{x^2}{6}+\frac{y^2}{10}& =& 1\end{array}$

The standard equation of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Compare the resultant equation with the standard equation of an ellipse.

Therefore, the equation $\frac{x^2}{6}+\frac{y^2}{10}=1$represents the equation of an ellipse with $a^2=6$and $b^2=10$.