Question

Evaluate by cursory inspection: $$ \begin{array}{l} \text { a. det }\left[\begin{array}{ccc} a & b & c \\ a+1 & b+1 & c+1 \\ a-1 & b-1 & c-1 \end{array}\right] \\ \text { b. det }\left[\begin{array}{ccc} a & b & c \\ a+b & 2 b & c+b \\ 2 & 2 & 2 \end{array}\right] \end{array} $$

Short answer

(a) Determinant is 0. (b) Direct calculation likely needed.

Step-by-step solution

Identify the Nature of the Matrix in (a)

In matrix (a), each row is a linear transformation of the others. For instance, the second row is each element of the first row increased by 1, and the third row is each element of the first row decreased by 1.

Evaluate the Determinant of Matrix (a)

Since the rows of the matrix are linear combinations or transformations of each other, this implies that the rows are linearly dependent. A matrix with linearly dependent rows has a determinant of zero. Therefore, \(\text{det} \left[ \begin{array}{ccc} a & b & c \ a+1 & b+1 & c+1 \ a-1 & b-1 & c-1 \end{array} \right] = 0\).

Identify the Nature of the Matrix in (b)

In matrix (b), the third row [2, 2, 2] is a constant row. Constant rows suggest a connection but still require checking if the other rows are linearly dependent.

Evaluate Linear Dependence in Matrix (b)

Multiply the first row by 2, resulting in [2a, 2b, 2c]. Subtract this new row from the second row [a+b, 2b, c+b], and examine the difference. If this subtraction results in a multiple of the third row [2, 2, 2], then the matrix is linearly dependent.

Determine the Nature of Subtraction in Matrix (b)

Calculate: \([a+b - 2a, 2b - 2b, c+b - 2c] = [-a+b, 0, -c+b]\). This is not a multiple of the third row [2, 2, 2], indicating linear independence, and thus it does not confirm zero determinant directly.

Conclusion on Matrix (b)

Given the unique makeup with independent rows, evaluate numerically or symbolically (through determinants properties) could be required, but there's no direct algebraic argument shortcut to answer, focus and test determinant directly if numeric or symbolic shortcut fails.

Key concepts

Linear Dependence

Linear dependence in matrices occurs when the rows (or columns) are linear combinations of each other. For instance, if you can express one row as a sum or multiple of other rows, the matrix is said to be linearly dependent. A practical example of this is the exercise you are working on. Here, in matrix (a), each row is closely related. The second row is formed by adding 1 to each element of the first row, and the third row is the result of subtracting 1 from each element of the first row.

This strong relationship means the rows are not independent, and due to this linear dependence, the determinant of the matrix becomes zero.

• If at least one row or column can be rendered as a linear combination of others, it indicates linear dependence.
• Linear dependence causes the matrix to lose rank, leading to a zero determinant.
• Remember: for determinants, zero signifies linear dependence among rows or columns.

Understanding linear dependence helps in simplifying determinant calculations, like immediately knowing that the determinant is zero in cases like matrix (a).

Matrix Transformations

Matrix transformations involve adjusting a matrix through operations like row and column manipulations. These transformations can significantly affect the nature and properties of the matrix, especially in terms of rank and determinant. An essential transformation related to this exercise is the concept of row operations forming linear combinations.

In matrix (b), while examining for linear dependence, transforming the first row by multiplying it by 2 and then subtracting resulted in a form that, at first glance, seemed independent of the third row ("Constant Row [2, 2, 2]").

• Row transformations can reveal or disguise dependencies between rows.
• The impact on rank and determinant varies with the operation. Simple transformations, like row addition or subtraction, maintain determinant value if properly balanced.
• It's crucial to thoroughly assess the roles of row transformations when determining linear dependencies.

Recognizing how transformations reveal dependencies or changes in matrix structure is key to understanding underlying matrix properties.

Determinant Properties

Determinant properties give us insight into how changes or conditions affect a matrix's determinant value. In linear algebra, several determinant rules are foundational, particularly focusing on outcomes like linear dependence and transformations.

A pivotal property exemplified in the exercise is that a matrix with linearly dependent rows has a determinant of zero. Matrix (a) in your exercise clearly shows this. Notably, any time rows (or columns) showcase a straightforward linear relationship like repetition or constant difference, the determinant plummets to zero.

• The determinant remains unchanged under specific transformations, barring rows swapping or multiplication.
• Understanding when determinants result in zero helps in quickly identifying redundant matrices for algebraic simplifications.
• Complex matrices demand checking dependencies and transformation effects to ascertain determinant values accurately.

Remember, the zero determinant serves as a clear indicator of dependency or systematic row repetition, crucial for evaluating both theoretical and practical problems in linear algebra.