Question

Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of RS 21. B buys 4 art books, 3 story books and 2 puzzle solving books for RS 60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays RS 10 more than B. Use matrix method to find the cost of each type of book.

Hint:

Always check your final answers by substituting them back into the original equations. For example: \( 5 + 8 + 8 = 21 \), which matches the first equation.

Answer 1

Concept:
A system of linear equations can be represented in matrix form as \[ AX = B \] and solved using \[ X = A^{-1}B, \] where \[ A^{-1} = \frac{1}{|A|}\operatorname{adj}(A). \]
Step 1: Form the linear equations
Let the cost of one art book be Rs. \(x\), one story book be Rs. \(y\), and one puzzle book be Rs. \(z\).
From student A: \[ x+y+z=21 \] From student B: \[ 4x+3y+2z=60 \] From student C: \[ 6x+2y+3z=70 \] 
Step 2: Represent the equations in matrix form
The system can be written as \[ \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & 2 \\ 6 & 2 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 21 \\ 60 \\ 70 \end{bmatrix} \] Thus, \[ A= \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & 2 \\ 6 & 2 & 3 \end{bmatrix} \] Now, \[ |A| = 1(9-4)-1(12-12)+1(8-18) \] \[ |A|=5-0-10=-5 \] Since \[ |A|\neq 0, \] the inverse \(A^{-1}\) exists. 
Step 3: Find the adjoint of \(A\)
The cofactor matrix is \[ \begin{bmatrix} 5 & 0 & -10 \\ -1 & -3 & 4 \\ -1 & 2 & -1 \end{bmatrix} \] Therefore, \[ \operatorname{adj}(A) = \begin{bmatrix} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{bmatrix} \] Hence, \[ A^{-1} = -\frac{1}{5} \begin{bmatrix} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{bmatrix} \] 
Step 4: Find \(X=A^{-1}B\)
\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = -\frac{1}{5} \begin{bmatrix} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{bmatrix} \begin{bmatrix} 21 \\ 60 \\ 70 \end{bmatrix} \] On multiplication, \[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = -\frac{1}{5} \begin{bmatrix} 5(21)-60-70 \\ -3(60)+2(70) \\ -10(21)+4(60)-70 \end{bmatrix} \] \[ = -\frac{1}{5} \begin{bmatrix} -25 \\ -40 \\ -40 \end{bmatrix} = \begin{bmatrix} 5 \\ 8 \\ 8 \end{bmatrix} \] 
Final Answer:
The cost of one art book is Rs. \(5\), one story book is Rs. \(8\), and one puzzle book is Rs. \(8\).