Class 12 Determinants · Case Study4 Marksmatrix_method_applications
Class 12 Maths · Determinants · Case Study #33
Sisters Reshma and Ritam have monthly incomes in the ratio 3:4 and expenditures in the ratio 5:7. Each saves ₹15,000 per month. Write incomes as 3x,4x and expenditures as 5y,7y.
Part (1) 1 Marks
Write the linear equations.
Answer: 3x−5y=15000; 4x−7y=15000.
- Savings equal income minus expenditure for each sister.
- Therefore 3x−5y=15000 and 4x−7y=15000.
Part (2) 1 Marks
Express the system as AX=B.
Answer: A=\(\begin{bmatrix}3 & -5\\4 & -7\end{bmatrix}\), X=[x,y]ᵀ, B=[15000,15000]ᵀ.
- Take coefficients in the order x,y: the rows are [3,−5] and [4,−7].
- The matrix form is \(\begin{bmatrix}3 & -5\\4 & -7\end{bmatrix}\)[x,y]ᵀ=[15000,15000]ᵀ.
Part (3) 2 Marks
Find A⁻¹.
Answer: \(\begin{bmatrix}7 & -5\\4 & -3\end{bmatrix}\)
- det(A)=3(−7)−(−5)4=−1≠0.
- For a 2×2 matrix, adj(A)=[[-7,5],[−4,3]], and A⁻¹=adj(A)/(−1).
- Hence A⁻¹=\(\begin{bmatrix}7 & -5\\4 & -3\end{bmatrix}\).
Part (4) 2 Marks
Alternatively, find C=A²−2I.
Answer: \(\begin{bmatrix}-13 & 20\\-16 & 27\end{bmatrix}\)
- Multiply A by itself to obtain A²=\(\begin{bmatrix}-11 & 20\\-16 & 29\end{bmatrix}\).
- Subtract 2I=[[2,0],[0,2]], leaving \(\begin{bmatrix}-13 & 20\\-16 & 27\end{bmatrix}\).
- Therefore C=\(\begin{bmatrix}-13 & 20\\-16 & 27\end{bmatrix}\).