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Q8:
Solve the system of linear equations, using matrix method: $2x – y = –2$, $3x + 4y = 3$
Solution :
Step 1: Matrix Representation of the System
The given system of linear equations is:
$2x - y = -2$
$3x + 4y = 3$
This system can be written in the matrix form $AX = B$, where:
$A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \end{pmatrix}$, and $B = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$
Step 2: Calculating the Determinant of Matrix A
To determine if the system has a unique solution, calculate $|A|$ [Per the Invertible Matrix Theorem]:
$|A| = \det\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} = (2 \times 4) - (-1 \times 3)$
$|A| = 8 + 3 = 11$
Since $|A| \neq 0$, the matrix $A$ is non-singular and therefore invertible. The unique solution exists and is given by $X = A^{-1}B$.
Step 3: Determining the Adjoint of Matrix A
For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the adjoint is $\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ [By definition of Adjoint Matrix].
$\text{adj}(A) = \begin{pmatrix} 4 & 1 \\ -3 & 2 \end{pmatrix}$
Step 4: Computing the Inverse Matrix $A^{-1}$
The inverse is calculated using the formula $A^{-1} = \frac{1}{|A|} \text{adj}(A)$:
$A^{-1} = \frac{1}{11} \begin{pmatrix} 4 & 1 \\ -3 & 2 \end{pmatrix}$
Step 5: Solving for X
Using $X = A^{-1}B$:
$X = \frac{1}{11} \begin{pmatrix} 4 & 1 \\ -3 & 2 \end{pmatrix} \begin{pmatrix} -2 \\ 3 \end{pmatrix}$
$X = \frac{1}{11} \begin{pmatrix} (4 \times -2) + (1 \times 3) \\ (-3 \times -2) + (2 \times 3) \end{pmatrix}$
$X = \frac{1}{11} \begin{pmatrix} -8 + 3 \\ 6 + 6 \end{pmatrix} = \frac{1}{11} \begin{pmatrix} -5 \\ 12 \end{pmatrix}$
$X = \begin{pmatrix} -5/11 \\ 12/11 \end{pmatrix}$
Final Solution: $x = -\frac{5}{11}, y = \frac{12}{11}$
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: $x + 3y = 5$, $2x + 6y = 8$
- 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: $5x + 2y = 4$, $7x + 3y = 5$
- 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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