# Applications of Determinants and Matrices

• The system of equations are said to be homogeneous system of equations if it is in the form
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0
(or)
a1x + b1y = 0
a2x + b2y = 0
• The system of equations are said to be non homogeneous equations, if it is in the form
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
(or)
a1x + b1y = d1
a2x + b2y = d2
• Solution of Homogeneous equations (Ax = 0)

• The system of equations are said to be consistent if it has unique solution (or) infinitely many solutions.
• The system of equations are said to be Inconsistent if it has no solution
• Solution of Non homogeneous equations (Ax = B)

### View the Topic in this video From 00:40 To 56:01

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• Homogeneous equations
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0 in matrix form is
\begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}
Ax = 0
• Non Homogeneous equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3 in matrix form is
\begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix}d_{1} \\ d_{2} \\ d_{3} \end{bmatrix}
A x = B
A = Coefficient matrix
x = solution matrix
• CRAMER'S RULE for solving of system of Non-homogeneous equations:
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
\Delta = \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}
\Delta_{1} = \begin{vmatrix}d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{vmatrix}
\Delta_{2} = \begin{vmatrix}a_{1} & d_{1} & c_{1} \\ a_{2} & d_{2} & c_{2} \\ a_{3} & d_{3} & c_{3} \end{vmatrix}
\Delta_{3} = \begin{vmatrix}a_{1} & b_{1} & d_{1} \\ a_{2} & b_{2} & d_{2} \\ a_{3} & b_{3} & d_{3} \end{vmatrix}
x=\frac{\Delta_1}{\Delta},y=\frac{\Delta_2}{\Delta},z=\frac{\Delta_3}{\Delta}
• MATRIX INVERSION METHOD for solving of Non-homogeneous equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
The System of equation in matrix form is
\begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix}d_{1} \\ d_{2} \\ d_{3} \end{bmatrix}
A x = B
⇒ x = A−1 B
\tt \Rightarrow x = \frac{1}{|A|}(adj A)(B)
• Echelon form of a matrix
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
Argument matrix → \begin{bmatrix}a_{1} & b_{1} & c_{1} & d_{1} \\ a_{2} & b_{2} & c_{2} & d_{2} \\ a_{3} & b_{3} & c_{3} & d_{3} \end{bmatrix}
↓ using Elementary Row operations
Echelon form of matrix → \begin{bmatrix}a_{1} & b_{1} & c_{1} & d_{1} \\ 0 & b_{2} & c_{2} & d_{2} \\ 0 & 0 & c_{3} & d_{3} \end{bmatrix}
Argument matrix → \begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \\ a_{4} & b_{4} & c_{4} \end{bmatrix}
↓ using Elementary column operations
Echelon form of matrix → \begin{bmatrix}a_{1} & 0 & 0 \\ a_{2} & b_{2} & 0 \\ a_{3} & b_{3} & c_{3} \\ a_{4} & b_{4} & c_{4} \end{bmatrix}
• Characteristic equation:
If A = \begin{bmatrix}1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix}, then the characteristic equation is |A − xI| = 0
\Rightarrow \begin{bmatrix}1-x & 1 & 0 \\ 1 & 2-x & 1 \\ 2 & 1 & -x \end{bmatrix} = 0
⇒ x3 − 3x2 − 1 = 0
The characteristic equation is
A3 − 3A2 − I = 0