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= [ a b c d ] 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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The way I&#039;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=      [                            a                          b                                      c                          d                      ]  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?

antidootnw · Accepted Answer

(1) A system of two linear equations in two variables can be written (in the standard way) as a matrix system   A            x      →        =            b      →       , where A is a   2  ×  2 matrix. Let&#039;s say its entries are  A  =      (                            a                          b                                      c                          d                      )    .The determinant of A is the value ad−bc. It&#039;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             b      →        is; it may not have a solution at all, depending on             b      →       . If   det  (  A  )  ≠  0, the system has a unique solution, no matter what             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&#039;re thinking of. That is a property of the determinant, but it would be a horrible definition of the determinant. It&#039;s hardly an obvious consequence of the definition of the determinant, either. But that&#039;s not to say you shouldn&#039;t keep in mind that it has this property.(3) See the end of (1).

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