Problem 9 - Algebra for JEE ADVANCED

Question

If \(A=\left[a_{i j}\right]_{4 \times 4}\), such that \(a_{i j}=\left\\{\begin{array}{ll}2, & \text { when } i=j \\ 0, & \text { when } i \neq j\end{array}\right.\), then \(\left\\{\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj} A))}{7}\right\\}\) is (where \(\\{\cdot\\}\) represents fractional part function) a. \(1 / 7\) b. \(2 / 7\) c. \(3 / 7\) d. none of these

Step-by-Step Solution

Answer

None of the options is correct; the answer is \(5/7\).

Step 1: Determine the Matrix Type

The matrix \(A\) is defined such that \(a_{ij}=2\) when \(i=j\) and \(a_{ij}=0\) when \(i eq j\). This means \(A\) is a 4x4 diagonal matrix with all diagonal elements equal to 2.

Step 2: Calculate the Determinant of A

For a diagonal matrix, the determinant is the product of its diagonal entries. Therefore, \(\det(A) = 2 \times 2 \times 2 \times 2 = 2^4 = 16\).

Step 3: Calculate the Adjugate of A

The adjugate of a diagonal matrix where all diagonal elements are equal to a constant \(k\), is also a diagonal matrix with each diagonal element equal to \(k^{n-1}\), where \(n\) is the matrix size. Here it is \(2^{3} = 8\). Hence, \(\text{adj}(A) = \begin{bmatrix} 8 & 0 & 0 & 0 \ 0 & 8 & 0 & 0 \ 0 & 0 & 8 & 0 \ 0 & 0 & 0 & 8 \end{bmatrix}\).

Step 4: Calculate the Determinant of Adjugate of A

As \(\text{adj}(A)\) is a diagonal matrix with all diagonal elements equal to 8, its determinant is \(8 \times 8 \times 8 \times 8 = 8^4 = 4096\).

Step 5: Calculate the Adjugate of Adjugate of A

Since \(\text{adj}(A)\) is a diagonal matrix, \(\text{adj}(\text{adj}(A))\) will also be a diagonal matrix with each diagonal element \(8^3 = 512\). Therefore, \(\text{adj}(\text{adj}(A)) = \begin{bmatrix} 512 & 0 & 0 & 0 \ 0 & 512 & 0 & 0 \ 0 & 0 & 512 & 0 \ 0 & 0 & 0 & 512 \end{bmatrix}\).

Step 6: Calculate the Determinant of Adjugate of Adjugate of A

The determinant of \(\text{adj}(\text{adj}(A))\) is \(512 \times 512 \times 512 \times 512 = 512^4\). Calculate \(512^4\), which is \(262144\).

Step 7: Divide Determinant by 7 and Find Fractional Part

Compute \(\frac{262144}{7}\), which results in approximately 37449.142857. The fractional part of this division is our previous answer should be `48`. `0.7142857...`, since it is a repeating factor that doesn’t fully reach one without the decimal division, you can double-check your pencil calculation.

Step 8: Determine the Correct Answer from Options

The fractional part calculated was \(\frac{5}{7}\), hence none of the given options \(\frac{1}{7}, \frac{2}{7}, \frac{3}{7}\) matches this value.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special kind of matrix. It has entries only along the diagonal from the top left to the bottom right. Everywhere else, the entries are zero. The general form of a diagonal matrix is:

• The elements along this diagonal can be any values, but all other elements are zero.
• For an n × n diagonal matrix, only n elements need to be specified since all others are known to be zero.
• In our exercise, matrix \( A \) is a diagonal matrix with values of 2 on the diagonal and zero everywhere else in a 4x4 matrix format.

This property makes it easy to calculate the determinant and perform operations like finding the adjugate or inverse. Working with diagonal matrices often simplifies matrix calculations.

Adjugate Matrix

The adjugate (or adjoint) matrix is a matrix that comes up when finding the inverse of another matrix. If you have a square matrix \( A \), the adjugate of \( A \), denoted as \( \operatorname{adj}(A) \), consists of the cofactors of \( A \) transposed.

• For diagonal matrices, finding the adjugate becomes straightforward. Each diagonal element in the adjugate is essentially the element to a power dependent on the size (n) of the matrix.
• For our matrix \( A \), with each diagonal element as 2 in a 4x4 form, the adjugate has elements of \( 2^{n-1} = 2^3 = 8 \).
• The resulting matrix is diagonal with entries 8 along the diagonal.

Understanding adjugates helps in computing inverses and determinants when dealing with square matrices.

Fractional Part Function

The fractional part function, denoted by \( \{ x \} \), refers to the part of the number that comes after the decimal point. In mathematical terms, for a real number \( x \), it is defined as:

• \( \{ x \} = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the floor function which gives the greatest integer less than or equal to \( x \).
• This function is especially useful in identifying patterns in real numbers and simplifying expressions that result in long decimal values.
• In our problem, after dividing the determinant of \( \operatorname{adj}(\operatorname{adj}(A)) \) by 7, we focus on the fractional part of the result.

The fractional part function often plays a role in mathematical problems involving periodicity and modular arithmetic.

Matrix Determinant Calculation

Calculating the determinant of a matrix provides information about the matrix including whether it is invertible. The determinant is a number that can be calculated from a square matrix. Here are some essentials about determinant calculation:

• For diagonal matrices, finding the determinant is simple; it's the product of the diagonal elements.
• As seen in the exercise, the determinant of matrix \( A \) was \( 2^4 = 16 \), which is achieved by multiplying the diagonal numbers together.
• Subsequent matrices, like the adjugate, maintain this property when they are also diagonal. Thus, \( \operatorname{det}(\operatorname{adj}(A)) = 8^4 \), and so on.

Understanding how to compute determinants helps in analyzing matrices, finding inverses, and solving systems of linear equations.