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Q4:
Examine the consistency of the system of equations: $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$
Solution :
Step 1: Matrix Representation of the System
The system of linear equations is given by:
- $x + y + z = 1$
- $2x + 3y + 2z = 2$
- $ax + ay + 2az = 4$
This can be represented in the form $AX = B$, where:
$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ a & a & 2a \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, $B = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}$
Step 2: Calculating the Determinant of Matrix A
To determine consistency, we must evaluate $|A|$ [Per Cramer’s Rule and the property of non-singular matrices].
$|A| = 1(3 \cdot 2a - 2 \cdot a) - 1(2 \cdot 2a - 2 \cdot a) + 1(2 \cdot a - 3 \cdot a)$
$|A| = 1(6a - 2a) - 1(4a - 2a) + 1(2a - 3a)$
$|A| = 4a - 2a - a = a$
Step 3: Evaluating Consistency Conditions
A system is consistent if $|A| \neq 0$ (unique solution) or if $|A| = 0$ and $(adj A)B = O$ (infinitely many solutions).
| Condition | Result |
|---|---|
| If $a \neq 0$ | $|A| \neq 0$, the system has a unique solution and is consistent. |
| If $a = 0$ | $|A| = 0$, we must check $(adj A)B$. |
Step 4: Analysis for $a = 0$
If $a = 0$, the matrix $A$ becomes:
$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 0 & 0 & 0 \end{pmatrix}$
The system of equations becomes:
- $x + y + z = 1$
- $2x + 3y + 2z = 2$
- $0x + 0y + 0z = 4 \implies 0 = 4$
Since $0 = 4$ is a mathematical contradiction [By the principle of logical consistency], there are no values of $x, y, z$ that satisfy the third equation.
Step 5: Final Conclusion
The system of equations is consistent for all $a \in \mathbb{R} \setminus \{0\}$. For $a = 0$, the system is inconsistent as it leads to a contradictory statement.
Final Solution: The system is consistent if $a \neq 0$ and inconsistent if $a = 0$.
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: $2x – y = 5$, $x + y = 4$
- Q3: Examine the consistency of the system of equations: $x + 3y = 5$, $2x + 6y = 8$
- 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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