Let us consider a system of n linear equations inn unknowns say `x_1, x_2, .x_3, ... , x_n` given as below :
`tt[(a_11x_1,+ , a_12x_2 , + ,a_13x_3 , ... , + ,a_{1n}x_n , = , b_1),(a_21x_1,+ , a_22x_2 , + ,a_23x_3 , ... , + ,a_{2n}x_n , = , b_2),(a_31x_1,+ , a_32x_2 , + ,a_33x_3 , ... , + ,a_{3n}x_n , = , b_3),(...,,...,,...,,,...,,...),(...,,...,,...,,,...,,...), (a_{n1}x_1,+ , a_{n2}x_2 , + ,a_{n3}x_3 , ... , + ,a_{n n}x_n , = , b_n)]}`
If `b_1= b_2 = b_3 = ... = b_n = 0,` then the system of Eq.(i) is called a system of homogeneous linear equations and if atleast one of `b_1, b_2, b_3 , ... , b_n` is non-zero, then it is called a system of non-homogeneous linear equation. We write the above system of Eq (i) in the matrix form as
`[(a_11,a_12,a_13,...,a_{1n}),(a_21,a_22,a_23,...,a_{2n}),(a_31,a_32,a_33,...,a_{3n}),(...,...,...,...,...),(...,...,...,...,...),(a_{n1},a_{n2},a_{n3},...,a_{3 n n})] [(x_1),(x_2),(x_3),(...),(...),(x_n)] [(b_1),(b_2),(b_3),(...),(...),(b_n)]`
`=> AX = B`
where
`A = [(a_11,a_12,a_13,...,a_{1n}),(a_21,a_22,a_23,...,a_{2n}),(a_31,a_32,a_33,...,a_{3n}),(...,...,...,...,...),(...,...,...,...,...),(a_{n1},a_{n2},a_{n3},...,a_{3 n n})] , X = [(x_1),(x_2),(x_3),(...),(...),(x_n)] text(and) B = [(b_1),(b_2),(b_3),(...),(...),(b_n)]`
Pre-multiplying Eq. (ii) by `A^(-1)` we get
`A^(-1)(AX) = A^(-1)B`
`=> (A^(-1)A) X = A^(-1)B => IX = A^(-1)B`
`=> X = A^(-1) B = ((adjA)B)/|A|`
`text(Types of Equations :)`
`(1)` `text(When system of equations is non-homogeneous)`
(i) If `| A | ne 0` , then the system of equations is consistent and has a umque solution given by `X= A^(-1) B`.
(ii) If `| A |= 0` and `(adjA) · B ne 0,` then the system of equations is inconsistent and has no solution.
(iii) If `| A |= 0` and `(adjA) · B = 0`, then the system of equations is consistent and has an infinite number of solutions.
`(2)` `text(When system of equations is homogeneous)`
(i) If `| A | ne 0,` then the system of equations has only trivial solution and it has one solution.
(ii) If `| A | = 0,` then the system of equations has non-trivial solution and it has infinite solutions.
(iii) If number of equations < number of unknowns, then it has non-trivial solution.
`text(Note)` Non-homogeneous linear equations can also be solved by Cramer's rule. this method has been discussed in the chapter on determinants.
Let us consider a system of n linear equations inn unknowns say `x_1, x_2, .x_3, ... , x_n` given as below :
`tt[(a_11x_1,+ , a_12x_2 , + ,a_13x_3 , ... , + ,a_{1n}x_n , = , b_1),(a_21x_1,+ , a_22x_2 , + ,a_23x_3 , ... , + ,a_{2n}x_n , = , b_2),(a_31x_1,+ , a_32x_2 , + ,a_33x_3 , ... , + ,a_{3n}x_n , = , b_3),(...,,...,,...,,,...,,...),(...,,...,,...,,,...,,...), (a_{n1}x_1,+ , a_{n2}x_2 , + ,a_{n3}x_3 , ... , + ,a_{n n}x_n , = , b_n)]}`
If `b_1= b_2 = b_3 = ... = b_n = 0,` then the system of Eq.(i) is called a system of homogeneous linear equations and if atleast one of `b_1, b_2, b_3 , ... , b_n` is non-zero, then it is called a system of non-homogeneous linear equation. We write the above system of Eq (i) in the matrix form as
`[(a_11,a_12,a_13,...,a_{1n}),(a_21,a_22,a_23,...,a_{2n}),(a_31,a_32,a_33,...,a_{3n}),(...,...,...,...,...),(...,...,...,...,...),(a_{n1},a_{n2},a_{n3},...,a_{3 n n})] [(x_1),(x_2),(x_3),(...),(...),(x_n)] [(b_1),(b_2),(b_3),(...),(...),(b_n)]`
`=> AX = B`
where
`A = [(a_11,a_12,a_13,...,a_{1n}),(a_21,a_22,a_23,...,a_{2n}),(a_31,a_32,a_33,...,a_{3n}),(...,...,...,...,...),(...,...,...,...,...),(a_{n1},a_{n2},a_{n3},...,a_{3 n n})] , X = [(x_1),(x_2),(x_3),(...),(...),(x_n)] text(and) B = [(b_1),(b_2),(b_3),(...),(...),(b_n)]`
Pre-multiplying Eq. (ii) by `A^(-1)` we get
`A^(-1)(AX) = A^(-1)B`
`=> (A^(-1)A) X = A^(-1)B => IX = A^(-1)B`
`=> X = A^(-1) B = ((adjA)B)/|A|`
`text(Types of Equations :)`
`(1)` `text(When system of equations is non-homogeneous)`
(i) If `| A | ne 0` , then the system of equations is consistent and has a umque solution given by `X= A^(-1) B`.
(ii) If `| A |= 0` and `(adjA) · B ne 0,` then the system of equations is inconsistent and has no solution.
(iii) If `| A |= 0` and `(adjA) · B = 0`, then the system of equations is consistent and has an infinite number of solutions.
`(2)` `text(When system of equations is homogeneous)`
(i) If `| A | ne 0,` then the system of equations has only trivial solution and it has one solution.
(ii) If `| A | = 0,` then the system of equations has non-trivial solution and it has infinite solutions.
(iii) If number of equations < number of unknowns, then it has non-trivial solution.
`text(Note)` Non-homogeneous linear equations can also be solved by Cramer's rule. this method has been discussed in the chapter on determinants.