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

$\left[\begin{array}{ccc}{\text{a}}_{\text{1}}& {\text{a}}_{\text{2}}& {\text{a}}_{\text{3}}\\ {\text{b}}_{\text{1}}& {\text{b}}_{\text{2}}& {\text{b}}_{\text{3}}\\ {\text{c}}_{\text{1}}& {\text{c}}_{\text{2}}& {\text{c}}_{\text{3}}\end{array}\right]$  $\left[\begin{array}{c}\text{x}\\ \text{y}\\ \text{z}\end{array}\right]$  = $\left[\begin{array}{c}{\text{k}}_{\text{1}}\\ {\text{k}}_{\text{2}}\\ {\text{k}}_{\text{3}}\end{array}\right]$        …(i)

Let A = $\left[\begin{array}{ccc}{\text{a}}_{\text{1}}& {\text{a}}_{\text{2}}& {\text{a}}_{\text{3}}\\ {\text{b}}_{\text{1}}& {\text{b}}_{\text{2}}& {\text{b}}_{\text{3}}\\ {\text{c}}_{\text{1}}& {\text{c}}_{\text{2}}& {\text{c}}_{\text{3}}\end{array}\right]$  ; X = $\left[\begin{array}{c}\text{x}\\ \text{y}\\ \text{z}\end{array}\right]$  and B = $\left[\begin{array}{c}{\text{k}}_{\text{1}}\\ {\text{k}}_{\text{2}}\\ {\text{k}}_{\text{3}}\end{array}\right]$

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.