Problem 5 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(P=\left[\begin{array}{lll}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{array}\right]\) and \(\mathrm{Q}=\left[\mathrm{q}_{\mathrm{ij}}\right]\) be two \(3 \times 3\) matrices such that \(\mathrm{Q}-\mathrm{P}^{5}=\mathrm{I}_{3}\). Then \(\frac{\mathrm{q}_{21}+\mathrm{q}_{31}}{\mathrm{q}_{32}}\) is equal to : [Jan. \(12,2019(\mathrm{I})]\) (a) 10 (b) 135 (c) 15 (d) 9

Step-by-Step Solution

Answer

The solution is (a) 10.

Step 1: Find the Matrix P^5

First, we need to calculate \( P^5 \). Matrix \( P \) is given by \( P = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} \).Since \( P \) is a lower triangular matrix with 1's on the diagonal, we know that the power of a triangular matrix is also triangular, and the diagonal elements are the same as the base matrix raised to that power. Calculating powers manually or verifying:\( P^2 = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 6 & 1 & 0 \ 27 & 6 & 1 \end{bmatrix} \)Proceed similarly to compute powers up to \( P^5 \). After computing sequential multiplications, you find:\( P^5 = \begin{bmatrix} 1 & 0 & 0 \ 15 & 1 & 0 \ 135 & 15 & 1 \end{bmatrix} \)

Step 2: Equation for Matrix Q

The problem states that \( Q - P^5 = I_3 \), where \( I_3 \) is the identity matrix:\( I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).Therefore, \( Q = P^5 + I_3 \).Given our \( P^5 \) from Step 1, we have:\( Q = \begin{bmatrix} 1 & 0 & 0 \ 15 & 1 & 0 \ 135 & 15 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \ 15 & 2 & 0 \ 135 & 15 & 2 \end{bmatrix} \).

Step 3: Calculate q_{21}, q_{31}, and q_{32}

From the matrix \( Q \), identify the necessary elements:- \( q_{21} \) is the element in row 2, column 1 which is 15.- \( q_{31} \) is the element in row 3, column 1 which is 135.- \( q_{32} \) is the element in row 3, column 2 which is 15.

Step 4: Solve for \( \frac{q_{21} + q_{31}}{q_{32}} \)

Substitute the values found in Step 3 into the expression: \( \frac{q_{21} + q_{31}}{q_{32}} = \frac{15 + 135}{15} \).Calculate the numerator:\( 15 + 135 = 150 \).Thus, the expression becomes:\( \frac{150}{15} = 10 \).

Key Concepts

Triangular Matrix

A triangular matrix is a special kind of square matrix where all the elements above or below the main diagonal are zero. There are two types of triangular matrices:

• Lower Triangular Matrix: All elements above the main diagonal are zero. An example would be matrix \( P \) from the exercise, which is lower triangular because all elements above its diagonal (i.e., non-zero elements) are zeros.
• Upper Triangular Matrix: All elements below the main diagonal are zero.

The diagonal of a triangular matrix can contain non-zero values. A distinctive feature of the triangular matrix in matrix exponentiation is that its powers remain triangular. This property helps simplify calculations when finding higher powers, like \( P^5 \) in the given problem. Since all elements not crucial to defining the structure are zero, the focus is mainly on the diagonal and the relevant side (lower or upper) of the matrix.

Identity Matrix

An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It is often denoted as \( I_n \), showing its dimensions, with \( n \) representing the number of rows and columns. In this problem, the identity matrix \( I_3 \) is a 3x3 matrix given by:\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]The identity matrix acts as the multiplicative identity in matrix algebra, similar to the number 1 in regular multiplication. When any matrix is multiplied by the identity matrix, it remains unchanged. This feature allows it to be a fundamental component in solving matrix equations.
The exercise uses this property in the equation \( Q - P^5 = I_3 \), leading to \( Q = P^5 + I_3 \). This relationship highlights how the identity matrix can help modify or adjust a matrix without changing its inherent structure.

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices to form a new matrix. Both matrices must have the same dimensions, which maintains structural consistency. For example, the matrices in this problem, \( P^5 \) and \( I_3 \), are 3x3, which allows them to add together without any issue.
To perform matrix addition, one would take an element from the first row and first column of the first matrix and add it to the element in the first row and first column of the second matrix, repeating this process for all matrix elements.
In the given exercise, we find matrix \( Q \) by adding \( P^5 \) and \( I_3 \):\[ Q = \begin{bmatrix} 1 & 0 & 0 \ 15 & 1 & 0 \ 135 & 15 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \ 15 & 2 & 0 \ 135 & 15 & 2 \end{bmatrix} \]Each element in \( Q \) is the sum of corresponding elements from \( P^5 \) and \( I_3 \).