Solution of a system of linear equations using inverse of a matrix
Consider the system of equations
a1x + a2y + a3z = k1
b1x + b2y + b3z = k2
c1x + c2y + c3z = k3
In matrix form, the system of equations can be expressed as
= …(i)
Let A = ; X = and B =
From (i), we have
AX = B
Case (i): A is a non-singular matrix. ⇒ A-1 exists.
We have AX = B
⇒ A-1 (AX) = A-1B
⇒ (A-1 A)X = A-1B (by associative property)
⇒ (I)X = A-1B ('.' A-1A = I , inverse property)
⇒ X = A-1B ⇒ X = A-1B ('.' IX = X , identity property)
This method of solving a system of linear equations is known as the matrix method.
Case (ii): A is a singular matrix.
⇒ |A| = 0
We have
X = A-1B ⇒ X = (adj A / Det A)B
Sub-case (i): (Adj A)B is a non-zero matrix.
In this case, the system of linear equations is inconsistent.
In other words, the system of linear equations has no solution.
Sub-case (ii): (Adj A)B is a zero matrix.
In this case, the system of linear equations has infinitely many solutions or no solution at all.
Solution of a system of linear equations using inverse of a matrix
Consider the system of equations
a1x + a2y + a3z = k1
b1x + b2y + b3z = k2
c1x + c2y + c3z = k3
In matrix form, the system of equations can be expressed as
= …(i)
Let A = ; X = and B =
From (i), we have
AX = B
Case (i): A is a non-singular matrix. ⇒ A-1 exists.
We have AX = B
⇒ A-1 (AX) = A-1B
⇒ (A-1 A)X = A-1B (by associative property)
⇒ (I)X = A-1B ('.' A-1A = I , inverse property)
⇒ X = A-1B ⇒ X = A-1B ('.' IX = X , identity property)
This method of solving a system of linear equations is known as the matrix method.
Case (ii): A is a singular matrix.
⇒ |A| = 0
We have
X = A-1B ⇒ X = (adj A / Det A)B
Sub-case (i): (Adj A)B is a non-zero matrix.
In this case, the system of linear equations is inconsistent.
In other words, the system of linear equations has no solution.
Sub-case (ii): (Adj A)B is a zero matrix.
In this case, the system of linear equations has infinitely many solutions or no solution at all.