Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) is an invertible matrix, then (A) \(\operatorname{adj} A^{\prime}=(\operatorname{adj} A)^{\prime}\) (B) \(\operatorname{adj} A^{\prime}=\operatorname{adj} A\) (C) \(\operatorname{adj} A^{\prime}=A^{\prime}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(\operatorname{adj} A^{\prime}=(\operatorname{adj} A)^{\prime}\).

Step 1: Define the Adjugate and Transpose

For a matrix \(A\), the adjugate is denoted as \(\operatorname{adj}(A)\). For any square matrix, the adjugate \(\operatorname{adj}(A)\) is the transpose of its cofactor matrix. The transpose of a matrix \(A\) is denoted by \(A'\) or \(A^T\).

Step 2: Use the Property of Adjugate and Transpose

The property we need to use is that for any invertible matrix \(A\), the adjugate of the transpose of \(A\), \(\operatorname{adj}(A')\), is the transpose of the adjugate of \(A\), which is \((\operatorname{adj}(A))'\). This is a fundamental property of adjugates and transposes.

Step 3: Verify Option (A)

Option (A) states \(\operatorname{adj} A^{\prime}=(\operatorname{adj} A)^{\prime}\). This matches exactly with the property that \(\operatorname{adj}(A') = (\operatorname{adj}(A))'\).

Step 4: Check Other Options

Option (B) suggests \(\operatorname{adj} A^{\prime} = \operatorname{adj} A\), which is not generally true because the adjugate of a transpose is not the adjugate itself. Likewise, Option (C) breaks down for similar reasons since \(\operatorname{adj} A^{\prime} = A^{\prime}\) implies that the adjugate is the matrix itself, which contradicts the properties of adjugates and transposes.

Step 5: Conclude With the Correct Option

Based on the verification of each possibility, the correct choice is Option (A), which accurately uses the mathematical property of the adjugate of the transpose.

Key Concepts

Transpose of a Matrix

A transpose of a matrix, often noted as \(A^T\) or \(A'\), is a new matrix derived by interchanging the rows and columns of the original matrix \(A\). This means if \(A\) is a matrix with elements \(a_{ij}\), then its transpose \(A^T\) will have elements \(a_{ji}\). Simply put, the element that was in the position (row \(i\), column \(j\)) moves to (row \(j\), column \(i\)).

This operation has some interesting properties:

• The transpose of a transpose returns the original matrix, i.e., \((A^T)^T = A\).
• The transpose of a sum of two matrices is the sum of their transposes, \((A + B)^T = A^T + B^T\).
• For a product of two matrices, the transpose meets these criteria \((AB)^T = B^TA^T\), where their order switches in the transposition process.

These properties make the transpose a crucial tool in matrix calculations, such as finding the adjugate or dealing with inverse matrices. They help simplify expressions and solve many problems in linear algebra.

Invertible Matrix Properties

An invertible matrix, also known as a non-singular or non-degenerate matrix, is one that has an inverse. We denote the inverse of a matrix \(A\) as \(A^{-1}\). An important property for a square matrix to be invertible is that its determinant, \(\det(A)\), must be non-zero.

Key properties of invertible matrices include:

• \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix that serves as a neutral element in matrix multiplication.
• The inverse of a transpose is the transpose of the inverse: \((A^T)^{-1} = (A^{-1})^T\).
• If a product of two matrices \(AB\) is invertible, then both \(A\) and \(B\) are individually invertible.

These properties are critical, as they pop up frequently when solving linear systems, eigenvalue problems, and much more. An invertible matrix guarantees unique solutions for systems of linear equations.

Matrix Cofactor

Cofactors are a central concept in calculating the adjugate and determinant of matrices. A cofactor of an element \(a_{ij}\) in a matrix is computed by:

• First finding the minor of the element, which is the determinant of the submatrix formed when the \(i\)-th row and \(j\)-th column are removed from the full matrix.
• Then multiplying the minor by \((-1)^{i+j}\). This accounts for the alternating signs across the matrix.

The matrix of cofactors is assembled by computing the cofactor for each element and arranging them in the matrix form.

Once we have the matrix of cofactors, its transpose is what we call the adjugate of the original matrix. The adjugate plays a vital role in determining the inverse of a matrix through the formula \(A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A)\) when \(\det(A) eq 0\).

Understanding cofactors aids immensely in grasping deeper matrix algebra concepts, especially when dealing with higher-level calculations and problem-solving in mathematics.