Problem 47 - Algebra for JEE ADVANCED

Question

If \(\boldsymbol{A}\) is a square matrix of order \(n\) such that \(|\operatorname{adj}(\operatorname{adj} A)|=|A|^{9}\), then the value of \(n\) can be \(\begin{array}{ll}\text { a. } 4 & \text { b. } 2\end{array}\) \(\begin{array}{ll}\text { c. either } 4 \text { or } 2 & \text { d. none of these }\end{array}\)

Step-by-Step Solution

Answer

n can only be 4.

Step 1: Understanding the property of adjugates

For a square matrix \(A\) of order \(n\), it satisfies \(|A|^{n-1} = |\text{adj}(A)|\). This adjugate property will be crucial in our calculations for the problem.

Step 2: Applying the adjugate property once

Given the problem statement, we have \(|\operatorname{adj}(\operatorname{adj}(A))| = |A|^9\). Using the adjugate property on \(\operatorname{adj}(A)\), we can express this as \(|\operatorname{adj}(A)|^{n-1} = |A|^9\).

Step 3: Using the adjugate formula again

We know \(|\operatorname{adj}(A)| = |A|^{n-1}\). Substituting this into the equation from the previous step gives us \((|A|^{n-1})^{n-1} = |A|^9\). This simplifies to \(|A|^{(n-1)^2} = |A|^9\).

Step 4: Equating powers of determinants

Assuming \(|A| eq 0\), we can equate the exponents: \((n-1)^2 = 9\). This implies \(n-1 = 3\) or \(n-1 = -3\).

Step 5: Finding possible values of n

From \(n-1 = 3\), we have \(n = 4\). From \(n-1 = -3\), we have \(n = -2\). Since \(n\) represents the order of a matrix, it must be a positive integer.

Step 6: Conclusion on the value of n

Since only \(n=4\) is a valid matrix order, the value of \(n\) can only be 4, considering \(n\) is a positive integer.

Key Concepts

Adjugate Matrix

An adjugate matrix, frequently referred to as the adjoint matrix, is a matrix that plays a significant role in the computation of matrix inverses and determinants. It is constructed from the cofactors of a given square matrix. To build the adjugate matrix of a matrix \( A \, \), take the following steps:

• Calculate the minor for each element of \( A \, \). A minor of an element is the determinant of the submatrix formed by deleting the row and column of that element.


• Find the cofactor for each element. The cofactor is calculated by multiplying the corresponding minor by \( (-1)^{i+j} \, \), where \( i \, \) and \( j \, \) are the row and column indices, respectively.


• Transpose the cofactor matrix, so rows become columns and columns become rows. This transposed matrix is the adjugate matrix.

The adjugate is vital in the formula for calculating the inverse of a matrix. If \( |A| \, \eq 0 \,\), then the inverse of \( A \, \) is given by \( A^{-1} = \frac{1}{|A|} \text{adj}(A) \,\).
Understanding adjugates helps appreciate how they are used to solve problems involving powers of determinants, as seen in this exercise.

Matrix Order

The order of a matrix, often highlighted when discussing determinants, adjugates, and other matrix operations, simply refers to the matrix's dimensions: its number of rows and columns. Specifically, in square matrices, which are the focus here, the order is the number of rows (or columns) since rows and columns are equal in number.

• A matrix of order \( n \, \) means that it has \( n \) rows and \( n \) columns, forming a \( n \times n \) square matrix.


• This order defines many properties of the matrix, including the complexity of its determinant.


• It's critical because certain operations and properties, such as finding inverses or adjugates, are typically defined only for square matrices.

In this exercise, understanding the matrix order is essential because the power relation given in terms of \( |A|^{n-1} \) stems directly from the order of the square matrix being discussed.

Determinant Properties

Determinants are scalar values that are peculiar yet fundamental to square matrices. They give insights into many properties, such as matrix invertibility and the volume transformation associated with the linear map defined by a matrix.

• The basic property is that if the determinant of a matrix \( A \, \), denoted as \( |A| \), is zero, then \( A \, \) is singular, meaning it does not have an inverse.


• The determinant has a multiplicative property: if \( A \, \) and \( B \, \) are matrices, then \( |AB| = |A| \cdot |B| \).


• In specific calculations, like in this exercise, properties help determine relations such as \( |\text{adj}(A)| = |A|^{n-1} \) and \( |\text{adj}(\text{adj}(A))| = |A|^{n(n-2)} \).

Understanding determinant properties enables solving complex algebraic problems involving powers and external structures like adjugates.

Square Matrix

A square matrix is a type of matrix where the number of rows is equal to the number of columns, which gives it a shape of a perfect square. These matrices are fundamental because the majority of determinant-related properties and operations like finding adjugates are defined specifically for square matrices.

• The structure of a square matrix allows us to perform operations like calculating determinants, inverses, and adjugates effectively.


• In linear algebra, many theorems and definitions hinge on square matrices due to their symmetry and defined size, such as the Cayley-Hamilton theorem and even eigenvalue theory.


• A \(n \times n \) square matrix offers a uniform way to discuss various crucial matrix operations.

Considering square matrices helps approach problems with a clear definition of size and allows for streamlined computation and application of various linear algebraic properties.