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Magazine Chapter 4: Problem 66
If \(A\) is an \(n \times n\) matrix, prove that \\[ \operatorname{det}(a d j A)=(\operatorname{det} A)^{n-1} \\]
Short Answer
Expert verified
\( \operatorname{det}(\operatorname{adj} A) = (\operatorname{det} A)^{n-1} \).
Step by step solution
01
Understanding the Adjugate Matrix
To begin with, the adjugate (or adjoint) of a matrix, denoted as \( \operatorname{adj} A \), is the transpose of the cofactor matrix of \( A \). If \( A \) is an \( n \times n \) matrix, then each element of the cofactor matrix is calculated using \( (n-1) \times (n-1) \) determinants of submatrices of \( A \).
02
Using the Adjugate and Determinant Relation
Recall that for any square matrix \( A \), the formula \( A \cdot \operatorname{adj} A = \operatorname{det} A \cdot I_n \) holds, where \( I_n \) is the \( n \times n \) identity matrix. This means that multiplying \( A \) by its adjugate results in a diagonal matrix which is scaled by the determinant of \( A \).
03
Computing the Determinant of Both Sides
Compute the determinant of both sides of the equation from the previous step: \( \operatorname{det}(A \cdot \operatorname{adj} A) = \operatorname{det}(\operatorname{det} A \cdot I_n) \).
04
Evaluating the Determinant of the Scalar Matrix
The determinant of a scalar multiple of an identity matrix, \( \operatorname{det}(\operatorname{det} A \cdot I_n) \), is \((\operatorname{det} A)^n\) because it's equivalent to \( (\operatorname{det} A)^n = \operatorname{det} A^n \).
05
Applying Mixed Determinant Properties
For matrices \( A \) and \( B \), \( \operatorname{det}(A \cdot B) = \operatorname{det}(A) \cdot \operatorname{det}(B) \). Therefore, \( \operatorname{det}(A \cdot \operatorname{adj} A) = \operatorname{det}(A) \cdot \operatorname{det}(\operatorname{adj} A) \).
06
Equating Expressions
From steps 3 and 5, we equate \( \operatorname{det}(A) \cdot \operatorname{det}(\operatorname{adj} A) \) with \( (\operatorname{det} A)^n \).
07
Solving for the Determinant of the Adjugate
Solve the equation \( \operatorname{det}(A) \cdot \operatorname{det}(\operatorname{adj} A) = (\operatorname{det} A)^n \) to find \( \operatorname{det}(\operatorname{adj} A) = (\operatorname{det} A)^{n-1} \). This confirms the expression for the determinant of the adjugate in terms of the determinant of \( A \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant
The determinant, often represented as \( \operatorname{det}(A) \), is a special number that can be calculated from a square matrix. This number gives us important information about the matrix:
- Area/Volume Interpretation: For a 2x2 matrix, the determinant gives the area of the parallelogram formed by the matrix's column vectors; for a 3x3 matrix, it gives the volume of the parallelepiped.
- Singularity: If the determinant of a matrix is zero, the matrix is considered singular, meaning it doesn't have an inverse.
- Scaling Factor: In transformations, the determinant acts as a factor, scaling areas or volumes.
Cofactor matrix
The cofactor matrix is derived from another matrix by calculating the cofactors of its elements. Understanding cofactors is key:
- Minor: For any element \( a_{ij} \) in a matrix, its minor is the determinant of the submatrix that remains after removing the element's row and column.
- Cofactor: The cofactor is then calculated by applying a sign pattern, \((-1)^{i+j}\), to the minor, where \( i \) and \( j \) are the row and column indices of the element. So, \( C_{ij} = (-1)^{i+j}M_{ij} \), where \( M_{ij} \) is the minor.
Identity matrix
The identity matrix is like the number 1 for matrices; multiplying any matrix by an identity matrix leaves the original matrix unchanged. It is denoted as \( I_n \) for an \( n \times n \) matrix and its properties include:
- Diagonal Elements: All diagonal elements are 1, and all off-diagonal elements are 0. For example, a 3x3 identity matrix looks like \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).
- Multiplication: For any square matrix \( A \) of the same size, \( A \cdot I_n = I_n \cdot A = A \).
- Inverse Relation: If a square matrix \( A \) has an inverse, then \( A \cdot A^{-1} = A^{-1} \cdot A = I \), where \( I \) is the identity matrix.
Square matrix
A square matrix is simply a matrix with the same number of rows and columns. Let's understand its features:
- Dimensions: A square matrix has dimensions \( n \times n \), meaning it has equal rows and columns.
- Determinant Defined: Only square matrices have determinants. This property makes them critical in discussing singularity, inversion, and volume scaling.
- Diagonal Properties: Elements that lie on a line from the top-left to the bottom-right are part of the main diagonal, an essential feature for identity matrices.
- Symmetric and Orthogonal: Some special square matrices, like symmetric and orthogonal matrices, have unique properties that are important in advanced topics.
