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Q1:
Examine the consistency of the system of equations: $x + 2y = 2$, $2x + 3y = 3$
Solution :
Step 1: Representing the System in Matrix Form
The given system of linear equations is:
$x + 2y = 2$
$2x + 3y = 3$
This system can be represented as the matrix equation $AX = B$, where:
$A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, and $B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$
[Per the matrix representation of linear systems $AX=B$]
Step 2: Calculating the Determinant of Matrix A
To determine the consistency of the system, we must evaluate the determinant $|A|$. A system is consistent if $|A| \neq 0$ (unique solution) or if $|A| = 0$ and $(adj A)B = O$ (infinitely many solutions).
$|A| = \det\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$
$|A| = (1 \times 3) - (2 \times 2)$
$|A| = 3 - 4 = -1$
[By the definition of the determinant of a $2 \times 2$ matrix: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$]
Step 3: Analytical Inference
Since $|A| = -1$, it follows that $|A| \neq 0$.
[By the Non-Singular Matrix Theorem: If the determinant of the coefficient matrix is non-zero, the matrix $A$ is invertible ($A^{-1}$ exists), and the system possesses a unique solution defined by $X = A^{-1}B$.]
Step 4: Visualizing the System Geometry
Step 5: Final Conclusion
Because $|A| \neq 0$, the matrix $A$ is non-singular. Consequently, the system of equations is consistent and has a unique solution.
Final Solution: The system of equations is consistent.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.5
- Q2: Examine the consistency of the system of equations: $2x – y = 5$, $x + y = 4$
- Q3: Examine the consistency of the system of equations: $x + 3y = 5$, $2x + 6y = 8$
- 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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