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Q13:
Solve the system of linear equations, using matrix method: $2x + 3y +3 z = 5$, $x – 2y + z = – 4$, $3x – y – 2z = 3$
Solution :
Step 1: Matrix Representation of the System
The system of linear equations is given by:
- $2x + 3y + 3z = 5$
- $x - 2y + z = -4$
- $3x - y - 2z = 3$
This system can be written in the form $AX = B$, where:
$A = \begin{pmatrix} 2 & 3 & 3 \\ 1 & -2 & 1 \\ 3 & -1 & -2 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, $B = \begin{pmatrix} 5 \\ -4 \\ 3 \end{pmatrix}$Step 2: Determining Invertibility via Determinant Analysis
We calculate the determinant $|A|$ to ensure the inverse $A^{-1}$ exists [Condition for unique solution: $|A| \neq 0$].
$|A| = 2((-2)(-2) - (1)(-1)) - 3((1)(-2) - (1)(3)) + 3((1)(-1) - (-2)(3))$ $|A| = 2(4 + 1) - 3(-2 - 3) + 3(-1 + 6)$ $|A| = 2(5) - 3(-5) + 3(5) = 10 + 15 + 15 = 40$Since $|A| = 40 \neq 0$, the matrix $A$ is non-singular, and the system has a unique solution $X = A^{-1}B$.
Step 3: Computing the Adjoint of Matrix A
The inverse is calculated as $A^{-1} = \frac{1}{|A|} \text{adj}(A)$. We find the cofactor matrix $C_{ij}$:
| $C_{11} = +((-2)(-2) - (-1)(1)) = 5$ | $C_{12} = -((1)(-2) - (3)(1)) = 5$ | $C_{13} = +((1)(-1) - (3)(-2)) = 5$ |
| $C_{21} = -((3)(-2) - (-1)(3)) = 3$ | $C_{22} = +((2)(-2) - (3)(3)) = -13$ | $C_{23} = -((2)(-1) - (3)(3)) = 11$ |
| $C_{31} = +((3)(1) - (-2)(3)) = 9$ | $C_{32} = -((2)(1) - (1)(3)) = 1$ | $C_{33} = +((2)(-2) - (1)(3)) = -7$ |
Transposing the cofactor matrix gives $\text{adj}(A)$:
$\text{adj}(A) = \begin{pmatrix} 5 & 3 & 9 \\ 5 & -13 & 1 \\ 5 & 11 & -7 \end{pmatrix}$Step 4: Calculating the Variable Vector X
Applying $X = \frac{1}{|A|} \text{adj}(A)B$:
$X = \frac{1}{40} \begin{pmatrix} 5 & 3 & 9 \\ 5 & -13 & 1 \\ 5 & 11 & -7 \end{pmatrix} \begin{pmatrix} 5 \\ -4 \\ 3 \end{pmatrix}$ $X = \frac{1}{40} \begin{pmatrix} (5)(5) + (3)(-4) + (9)(3) \\ (5)(5) + (-13)(-4) + (1)(3) \\ (5)(5) + (11)(-4) + (-7)(3) \end{pmatrix}$ $X = \frac{1}{40} \begin{pmatrix} 25 - 12 + 27 \\ 25 + 52 + 3 \\ 25 - 44 - 21 \end{pmatrix} = \frac{1}{40} \begin{pmatrix} 40 \\ 80 \\ -40 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$Visual Representation of Matrix Transformation
Final Solution: x = 1, y = 2, z = -1
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: $5x + 2y = 4$, $7x + 3y = 5$
- 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: $x – y + z = 4$, $2x + y – 3z = 0$, $x + y + z = 2$
- 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
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