The way I've been introduced to determinants is that if there is a system of two linear equations then we can represent the coefficients of the variables and the constants in the form of a matrix.
Now if we plot the matrices on the coordinate system then we will get a parallelogram and if we calculate the area of the parallelogram then we will get the determinant of the given matrix.
For eg if A is the matrix then its determinant will be:
$ad-cb$.
i.e. |A|= $ad-cb$.
if A=$\begin{bmatrix}a & b\\c & d\end{bmatrix}$
Now the questions I want to ask:
1)What is a determinant actually what does it tells us about a system of equations?
2)The area found by the formula $ad-cb$, how is it telling us a determinant? Basically how the area of parallelogram telling the value of determinant?
3)In my book its given that: system of equations has a unique solution or not is determined by the number of ab-cd.What does this mean?
Best Answer
(1) A system of two linear equations in two variables can be written (in the standard way) as a matrix system $A \vec x = \vec b$, where $A$ is a $2\times 2$ matrix. Let's say its entries are $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$ The determinant of $A$ is the value $ad-bc$. It's (mostly) only important whether this value is zero or nonzero. If $\det(A) = 0$, the system of equations does not have a unique solution (meaning it has either no or more than one solution), no matter what $\vec b$ is; it may not have a solution at all, depending on $\vec b$. If $\det(A) \ne 0$, the system has a unique solution, no matter what $\vec b$ is. You can prove this quite easily in the case of $2\times 2$ matrices.
(2) The determinant is the value $ad-bc$. It happens to also give you the (signed) area of the parallelogram you're thinking of. That is a property of the determinant, but it would be a horrible definition of the determinant. It's hardly an obvious consequence of the definition of the determinant, either (unless you really only care about $2\times 2$ matrices, in which case, see @J.G.'s answer). But that's not to say you shouldn't keep in mind that it has this property.
(3) See the end of (1).