Matrix based System of Linear Equations Class 12 Mathematics Notes has been updated according to the latest syllabus of 2080. It means the solutions of Matrix based System of Linear Equations chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 5
Matrix based System of Linear Equations
We come across equations, when we try to solve some problems in mathematics. These equations may be of one or more variables. The solutions of the equations give the solutions of the problems.
So it is quite natural that we should have knowledge of different methods of solving equations. The methods, we consider here, are row equivalent matrix method, Cramer’s rule or Determinant method and Inverse matrix method.
System of Linear Equations
A simple example of a linear equation in two variables x and y is 3x – y = 5. It is evident that solutions of this equation are ordered pairs such as (2, 1), (3, 4), (4, 7), etc. That is, each of the ordered pairs (2, 1), (3, 4), (4, 7), etc. satisfies the equation. For instance 3. [3] – [4] = 5 (true).
Matrix Based System of Linear Equations Notes PDF
This PDF will provide the solutions of every question from the 1st exercise of class 12 System of Linear Equations. If you want the solutions of other exercises then you can select the exercise from the button given above.
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AdminCase 1: Intersecting Lines
The graphs of the first pair 3x – y = 5 and x + y = 7 intersect at the point (3, 4) This means that both equations have the common solution x = 3 Y = 4. Although there is infinite number of solution sets which separately satisfy these equations, the solution x = 3 y = 4 is the only solution which satisfies both of them at the same time. This is why such a system of equations is called simultaneous equations.
Two linear equations in two variables which have at least one solution in common are said to be consistent. If consistent linear equations have just one solution in common, they are said to be independent. The system of equations x + y = 7 and 3x – y = 5 are not only consistent but also independent.
Case II: Parallel Lines
The graphs of the second pair x – y = 2 and x – y = 5 are parallel lines, that is they never meet. No set of values of x and y which satisfies one equation will satisfy the other equation. In other words, they have no common solution.
These equations are inconsistent and independent. We can see why they are so called. For one equation states that the difference of two numbers x and y is 2 while the other states that their difference is 5, and two such numbers whose difference is both 2 and 5 at the same time do not exist.
Case III: Coincident Lines
The graphs of the pair of equations 3x – y = 5 and 6x – 2y = 10 coincide (i.e. one fits exactly on the other). Obviously, these equations are consistent, that is, a set of values of x and y which satisfies the first equation satisfies the second also.
Moreover, these equations are dependent, because one of them can be obtained from the other just by multiplying both sides by a constant. In the above case, the second can be obtained from the first by multiplying both sides of the first by 2.
From the above discussion, a system of linear equations;
- a1x + b1y = c1
- a2x + b2y = c2
where x and y are two variables and the rest are all constants, may be classified in the following way:
- The system is consistent and independent if the equations have exactly one solution, i.e. if their graphs intersect in exactly one point.
- The system is inconsistent and independent if the equations have no solution common to both, i.e. if the graphs of the equations do not intersect.
- The system is consistent and dependent if every solution of one of the equations is also a solution of the other i.e. if the graphs of the equations coincide.
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