# 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=[acbd] 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 un

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:
if A=$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
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?
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antidootnw
(1) A system of two linear equations in two variables can be written (in the standard way) as a matrix system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ , where A is a $2×2$ matrix. Let's say its entries are
$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right).$
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 $\stackrel{\to }{b}$ is; it may not have a solution at all, depending on $\stackrel{\to }{b}$ . If $det\left(A\right)\ne 0$, the system has a unique solution, no matter what $\stackrel{\to }{b}$ is. You can prove this quite easily in the case of $2×2$ 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. But that's not to say you shouldn't keep in mind that it has this property.
(3) See the end of (1).