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Q3:
Examine the consistency of the system of equations: $x + 3y = 5$, $2x + 6y = 8$
Solution :
Step 1: Representing the System in Matrix Form
A system of linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ can be expressed as $AX = B$, where:
$A = \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, and $B = \begin{bmatrix} 5 \\ 8 \end{bmatrix}$
Step 2: Calculating the Determinant of Matrix $A$
To determine consistency, we must evaluate $\det(A)$ or $|A|$. [Per the property of consistency: If $|A| \neq 0$, the system is consistent with a unique solution. If $|A| = 0$, the system may be inconsistent or have infinitely many solutions.]
$|A| = \begin{vmatrix} 1 & 3 \\ 2 & 6 \end{vmatrix} = (1 \times 6) - (3 \times 2)$
$|A| = 6 - 6 = 0$
Step 3: Evaluating Consistency via the Adjoint Matrix Method
Since $|A| = 0$, we must compute the matrix $adj(A) \times B$. [Condition for consistency when $|A| = 0$: If $(adj A)B = O$, the system is consistent (infinitely many solutions); if $(adj A)B \neq O$, the system is inconsistent.]
For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $adj(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
$adj(A) = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}$
Now, calculate $(adj A)B$:
$(adj A)B = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} 5 \\ 8 \end{bmatrix} = \begin{bmatrix} (6 \times 5) + (-3 \times 8) \\ (-2 \times 5) + (1 \times 8) \end{bmatrix}$
$(adj A)B = \begin{bmatrix} 30 - 24 \\ -10 + 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}$
Step 4: Conclusion of Consistency
Since $(adj A)B \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, the system of equations has no solution. Geometrically, the two lines represented by the equations are parallel and non-coincident.
Final Solution: The system of equations is inconsistent.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.5
- Q1: Examine the consistency of the system of equations: $x + 2y = 2$, $2x + 3y = 3$
- Q2: Examine the consistency of the system of equations: $2x – y = 5$, $x + y = 4$
- Q4: Examine the consistency of the system of equations: $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$
- Q5: Examine the consistency of the system of equations: $3x–y – 2z = 2$, $2y – z = –1$, $3x – 5y = 3$
- Q6: Examine the consistency of the system of equations: $5x – y + 4z = 5$, $2x + 3y + 5z = 2$, $5x – 2y + 6z = –1$
- Q7: Solve the system of linear equations, using matrix method: $5x + 2y = 4$, $7x + 3y = 5$
- Q8: Solve the system of linear equations, using matrix method: $2x – y = –2$, $3x + 4y = 3$
- Q9: Solve the system of linear equations, using matrix method: $4x – 3y = 3$, $3x – 5y = 7$
- Q10: Solve the system of linear equations, using matrix method: $5x + 2y = 3$, $3x + 2y = 5$
- Q11: Solve the system of linear equations, using matrix method: $2x + y + z = 1$, $x – 2y – z = \frac{3}{2}$, $3y – 5z = 9$
- Q12: Solve the system of linear equations, using matrix method: $x – y + z = 4$, $2x + y – 3z = 0$, $x + y + z = 2$
- Q13: Solve the system of linear equations, using matrix method: $2x + 3y +3 z = 5$, $x – 2y + z = – 4$, $3x – y – 2z = 3$
- Q14: Solve the system of linear equations, using matrix method: $x – y + 2z = 7$, $3x + 4y – 5z = – 5$, $2x – y + 3z = 12$
- Q15: If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$ solve the system of equations $2x – 3y + 5z = 11$, $3x + 2y – 4z = – 5$, $x + y – 2z = – 3$
- Q16: The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is ` 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is ` 70. Find cost of each item per kg by matrix method.
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
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