Question

By computing the trace, determinant, and rank, show that \(A\) and \(B\) are not similar in each case. a. \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right]\) b. \(A=\left[\begin{array}{rr}3 & 1 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}2 & 1 \\ 1 & -1\end{array}\right], B=\left[\begin{array}{rr}3 & 0 \\ 1 & -1\end{array}\right]\) d. \(A=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right], B=\left[\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right]\) e. \(A=\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right], B=\left[\begin{array}{rrr}1 & -2 & 1 \\ -2 & 4 & -2 \\\ -3 & 6 & -3\end{array}\right]\) f. \(A=\left[\begin{array}{rrr}1 & 2 & -3 \\ 1 & -1 & 2 \\ 0 & 3 & -5\end{array}\right], B=\left[\begin{array}{rrr}-2 & 1 & 3 \\ 6 & -3 & -9 \\\ 0 & 0 & 0\end{array}\right]\)

Short answer

Matrices \( A \) and \( B \) are not similar in all cases.

Step-by-step solution

Introduction to Matrix Similarity

Two matrices are considered similar if there is an invertible matrix \(P\) such that \(A = PBP^{-1}\). For similarity, matrices must have the same trace, determinant, and rank.

Part a: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). For \(A\), \( \text{trace}(A) = 1 + 1 = 2\). For \(B\), \( \text{trace}(B) = 1 + 1 = 2\). The traces are equal.

Part a: Determinant Comparison

Calculate the determinant of matrices \(A\) and \(B\). \( \det(A) = 1(1) - 2(2) = 1 - 4 = -3 \). \( \det(B) = 1(1) - 1(-1) = 1 + 1 = 2 \). The determinants are different, so \(A\) and \(B\) are not similar.

Part b: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). \( \text{trace}(A) = 3 + (-1) = 2 \).\( \text{trace}(B) = 1 + 1 = 2 \).The traces are equal.

Part b: Determinant Comparison

Calculate the determinant of matrices \(A\) and \(B\).\( \det(A) = 3(-1) - 1(2) = -3 - 2 = -5\).\( \det(B) = 1(1) - 1(2) = 1 - 2 = -1\).The determinants are different, so \(A\) and \(B\) are not similar.

Part c: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). \( \text{trace}(A) = 2 + (-1) = 1 \).\( \text{trace}(B) = 3 + (-1) = 2 \).The traces are different, so \(A\) and \(B\) are not similar.

Part d: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). \( \text{trace}(A) = 3 + 2 = 5 \).\( \text{trace}(B) = 2 + 2 = 4 \).The traces are different, so \(A\) and \(B\) are not similar.

Part e: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). \( \text{trace}(A) = 2 + 0 + 0 = 2 \).\( \text{trace}(B) = 1 + 4 - 3 = 2 \).The traces are equal.

Part e: Rank Comparison

Calculate the ranks of matrices \(A\) and \(B\). \(A\) is full rank (nonsingular with rank 3), but \(B\) is rank 1 (due to dependent rows).Ranks differ, so \(A\) and \(B\) are not similar.

Part f: Trace Comparison

Calculate the trace of matrices \(A\) and \(B\). \( \text{trace}(A) = 1 + (-1) - 5 = -5 \).\( \text{trace}(B) = -2 - 3 = -5 \).The traces are equal.

Part f: Determinant Comparison

Calculate the determinant of matrices \(A\) and \(B\). For \(A\), use cofactor expansion:\[ \det(A) = 1(-1 imes (-5) - 2 imes 3) - 2(1 imes -5 - 2 imes 0) + (-3)(1 imes 3 - 0) = 1(5 - 6) - 2(-5) - 3 \times 3 = -1 + 10 - 9 = 0 \]\(B\) is rank deficient (one row is a multiple of another, implying \(\text{det}(B) = 0\)).Determinants are equal.

Part f: Rank Comparison

Calculate the ranks of matrices \(A\) and \(B\). \(A\) has full rank (rank 3), whereas \(B\) only has rank 2.Ranks differ, so \(A\) and \(B\) are not similar.

Key concepts

Trace of a Matrix

The trace of a matrix is one of the simplest numerical properties of a square matrix. It is defined as the sum of the elements on the main diagonal of the matrix. For example, if you have a matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]then the trace of matrix \(A\) is calculated as \( \text{trace}(A) = a + d \).
Understanding the trace is crucial when comparing two matrices for similarity. For two matrices to be similar, they must have the same trace.

• This is because similar matrices represent the same linear transformation but in different bases, resulting in the same sum for the main diagonal elements in all their equivalent forms.
• Thus, if the traces of two matrices are different, they cannot be similar.

For students, calculating the trace is often the first step in checking matrix similarity because it provides immediate insight into whether further checks (like determinants or ranks) are needed.

Determinant of a Matrix

The determinant is another fundamental property of a matrix, providing valuable information about the matrix's behavior and characteristics. It is particularly important for square matrices, where it indicates whether the matrix is invertible (non-zero determinant) or singular (zero determinant).
The determinant can be visualized as a scaling factor for the transformation represented by the matrix. A non-zero determinant suggests the preservation of dimensions and directions, while a zero determinant implies transformation leads to collapse into a lower dimension, such as lines or points.

• In 2x2 matrices, the determinant is calculated as: \[ \det\left( \begin{bmatrix} a & b \ c & d \end{bmatrix} \right) = ad - bc \]
• For similarity, two matrices must not only have the same rank and trace but also the same determinant.
• By examining determinants, one can rule out similarity when particularly different determinants are found, showcasing different scaling properties of corresponding transformations.

This measurement hints at geometric and algebraic properties, making it a crucial tool in linear algebra.

Rank of a Matrix

The rank of a matrix is its primary indicator of dimension in the vector space. It tells us the maximum number of linearly independent column vectors in the matrix, or equivalently, the number of linearly independent row vectors.
Rank essentially gives the insight into the span of the column or row vectors within the matrix, indicating the number of dimensions that are effectively used in your system.

• For matrix similarity, having the same rank is necessary as it verifies whether both matrices fill the same dimension of space, utilizing the same number of vectors for full rank representation.
• Diverging ranks between matrices mean different levels of independency and subspace spans, immediately ruling out similarity.

Calculating rank can involve reducing a matrix to its row echelon form or its reduced row echelon form.
Understanding the rank helps indeed in identifying whether a matrix maintains the capacity to account for each dimension of space independently. This attribute ensures transformations preserve volume and dimension, aligning closely with the concept of similarity among matrices.