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Q2:
Examine the consistency of the system of equations: $2x – y = 5$, $x + y = 4$
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 in the matrix equation form $AX = B$, where:
$A = \begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \end{pmatrix}$, and $B = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$
[Per the matrix representation of linear systems $AX = B$].Step 2: Calculating the Determinant of Matrix A
To determine consistency, we must evaluate the determinant $|A|$:
$|A| = \det \begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}$
$|A| = (2)(1) - (-1)(1)$ [By the definition of a $2 \times 2$ determinant: $ad - bc$]
$|A| = 2 + 1 = 3$
Step 3: Analyzing Consistency Criteria
A system $AX = B$ is consistent if $|A| \neq 0$ or if $|A| = 0$ and $(\text{adj } A)B = O$.
- Since $|A| = 3$, and $3 \neq 0$, the matrix $A$ is non-singular.
- A non-singular matrix implies the existence of a unique solution given by $X = A^{-1}B$.
Step 4: Geometric Representation (Visual Verification)
Step 5: Theoretical Conclusion
Because the determinant of the coefficient matrix is non-zero ($|A| = 3 \neq 0$), the system is algebraically guaranteed to possess a unique solution. Consequently, the system of equations is classified as consistent.
Final Solution: The system of equations is consistent as $|A| = 3 \neq 0$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.5
- Q1: Examine the consistency of the system of equations: $x + 2y = 2$, $2x + 3y = 3$
- 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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