 Hi and welcome to the session. Today we will learn about applications of determinants and matrices. First of all, let us learn what is a consistent system of linear equations. A system of equations is said to be consistent if its solution one or more exists. Now let us see what is an inconsistent system of linear equations. System of equations is said to be inconsistent if its solution does not exist. Now let us move on to the topic solution of system of linear equations using inverse of a matrix. System of equations a1x plus b1y plus c1z equal to d1, a2x plus b2y plus c2z equal to d2, a3x plus b3y plus c3z equal to d3. For this system of equations, matrix a is the matrix of the coefficients of the variables x, y and z in all the three equations. Matrix x is a column matrix of the variables x, y and z and matrix b is the column matrix of d1, d2 and d3. So the given system of equations is given by ax is equal to b. So here for this we have two cases. Case one is a is non-singular matrix. Then this implies that inverse of a exists. So the solution that is x will be given by a inverse b. Now this matrix equation provides unique solution for the given system of equations as the inverse of a matrix is unique. Also this method is known as matrix method. If a is a singular matrix then we will get determinant of a equal to 0. So in this case we will calculate a joint of a into b. A joint of a into b will not be equal to o where o is a zero matrix. Then the solution does not exist. And the system of equations is called inconsistent. A joint of a into b is equal to o where o is a zero matrix. Then system of equations may be either consistent or inconsistent according as the system of equations have either infinitely many solutions or no solution. So let's take one example. Here we need to solve the given system of linear equations. So here these are the matrices a, x and b and the given system of linear equations is written as a, x is equal to b. Now first of all let us find determinant of a which is equal to 1 and this is not equal to 0. So this implies the matrix a is non-singular that means system of linear equations have a unique solution. So let's find out a inverse that will be equal to 1 upon determinant a that is 1 into a joint of a which will be 3 minus 7 minus 2 5. Therefore x is equal to a inverse b which will be equal to 2 minus 3 and this is equal to x y that means x is equal to 2 and y is equal to minus 3. So this is the required solution for the given system of equations. We'll take one more example. Here again we are given a system of linear equations which we need to solve. So these are the matrices a, x and b. Now let's find determinant of a and that will be equal to 0. So this implies a is a singular matrix. Now we will calculate a joint of a into b and this will be equal to a column matrix that is 40 and minus 16 which is not equal to 0. Now from here we already know that if a joint of a into b is not equal to a 0 matrix then the solution does not exist and the system of equations is inconsistent. So here we have that solution does not exist and thus system of equations is inconsistent. So with this we finish this session. Hope you first have understood all the concepts. Goodbye, take care and have a nice day.