Question

Show that $$ \text { det }\left[\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_{1} \\ 0 & 0 & \cdots & a_{2} & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & a_{n-1} & \cdots & * & * \\ a_{n} & \+ & \cdots & \+ & + \end{array}\right]=(-1)^{k} a_{1} a_{2} \cdots a_{n} $$ where either \(n=2 k\) or \(n=2 k+1,\) and \(+\) -entries are artitrary.

Short answer

The determinant is the product \((-1)^k a_1 a_2 \cdots a_n\).

Step-by-step solution

Understand the Matrix Structure

The matrix given is an upper triangular matrix with potentially arbitrary entries marked by "+" in the last row. In a purely diagonal manner, each row from top to bottom except the last one has a single non-zero entry from the right (diagonal-wise and progressing upwards) along with zero entries elsewhere.

Identify the Non-Zero Elements

Observing the placement of non-zero elements, the main diagonal consists of the elements: - First non-zero element at the last column in the first row is \( a_1 \).- Second non-zero element at the second last column in the second row is \( a_2 \).- This pattern continues down to \( a_n \) in the last row, first column.These conform to the anti-diagonal form of products that lead to the determinant.

Evaluate the Determinant of an Upper Triangular Matrix

By the properties of determinants, the determinant of an upper triangular matrix is the product of its diagonal elements. The matrix given can be seen as having a skew diagonal forming a chain of non-zero elements which appear as diagonal entries in an upper triangular reform. Hence, the calculation involves directly multiplying these elements.

Calculate the Determinant Value

The matrix has precisely one invertible path of non-zero elements along this skew diagonal. So, the determinant becomes the product of elements: \[ a_1 imes a_2 imes ext{...} imes a_n \] Since each \(-1\) contributes a factor for every swap needed to order the indices along this path, the factor \((-1)^k\) accommodates the required eigenvalue sign adjustment by indicating the number of such swaps.

Confirm the Exponent Sign for (-1)

Reading the pattern in the placement of \( a_i \), it's essential to confirm the occurrences of row swaps by forming a path count, which is a function of the even or odd position of \( n \), ensuring the correct exponent value \((-1)^k\). This results from paths forming feasible skew (reversed) diagonal connect.

Key concepts

Upper Triangular Matrix

An upper triangular matrix is one where all the elements below the main diagonal are zero. In the context of our exercise, the matrix is structured such that it has a specific diagonal arrangement with some arbitrary elements in the last row. This type of matrix simplifies certain matrix operations, especially the calculation of determinants.

For an upper triangular matrix, the determinant is simply the product of the diagonal elements. In the given problem, the matrix is skewed such that non-zero elements form a 'chain' or a diagonal-like setup, not directly apparent as the main diagonal. However, the pattern of these elements follows a determinable chain, ensuring that despite its uncommon appearance, it can still be handled via properties of upper triangular matrices. Thus, finding the determinant involves multiplying these unique diagonal chain elements.

Understanding the structure of this matrix helps reduce computational complexity, allowing us to focus on these key non-zero elements and their arrangement to finding the determinant easily.

Matrix Diagonalization

Matrix diagonalization is a process where a matrix is expressed as a product of a matrix of its eigenvectors, a diagonal matrix of its eigenvalues, and the inverse of the eigenvector matrix. The diagonal matrix is composed only of the eigenvalues of the original matrix on its main diagonal.

In our exercise's context, the matrix seems intricate due to the displaced structure of its non-zero elements. However, it echoes the diagonalization strategy where recognizing the chain of determinant-relevant elements plays a crucial role, akin to finding eigenvalues in diagonalization.

This approach is similar to forming a usable diagonal matrix through inherent properties, aiding us in determining the determinant simply. Recognizing these patterns helps us economize calculations as we spot eigen-like patterns that define how this complex structure behaves mathematically.

Eigenvalue Analysis

Eigenvalue analysis involves exploring the properties of matrices that aid in various calculations, including diagonalization and determinant computation. Eigenvalues are intrinsic to this analysis, typically found by solving the characteristic equation of a matrix, often leading to substantial simplifications in algebraic contexts.

The exercise alludes to eigenvalue properties by requiring determinant calculation via its non-standard diagonal pattern. Though it doesn't directly present as an eigenvalue problem, the argument for (-1)^k suggests a resemblance to eigenconcepts. The swap effect resembles eigenvector transformations, which alter the orientation and possibly the value of system properties.

This type of analysis reinforces the structures seen here and in diagonal matrices, solving intricate problems more efficiently. Recognizing such properties in non-standard formats expands our toolset in linear algebra, making even skewed matrices predictable in behavior and allowing for computation of determinants and other critical matrix attributes.