Problem 1 - Algebra and Geometry

Question

Show that if \(A\) is a square matrix, then \(A A^{t}\) is symmetric. Choose any \(2 \times 2\) matrix and verify this directly.

Step-by-Step Solution

Answer

The matrix \(A A^{t}\) is symmetric, as both theoretical demonstration and computation with a \(2\times2\) example show.

Step 1: Understanding the Symmetric Matrix

A matrix is symmetric if it is equal to its own transpose, that is, for a matrix \(B\), \(B = B^{t}\). Our goal is to show that \(A A^{t}\) is symmetric for any square matrix \(A\).

Step 2: Calculate the Transpose of the Product

To show that \(A A^{t}\) is symmetric, we need to demonstrate that \((A A^{t})^{t} = A A^{t}\). We know the transpose of a product of matrices is the reverse product of their transposes. Therefore, \((A A^{t})^{t} = (A^{t})^{t} A^{t} = A A^{t}\). This confirms the product is symmetric.

Step 3: Verification with a 2x2 Matrix: Choose the Matrix

Let's choose a specific \(2 \times 2\) matrix \(A\) and show that \(A A^{t}\) is symmetric.Let \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\).

Step 4: Calculate the Transpose of the Example Matrix

The transpose of \(A\), denoted \(A^{t}\), is obtained by swapping rows and columns:\[ A^{t} = \begin{pmatrix} a & c \ b & d \end{pmatrix} \]

Step 5: Compute the Product \(A A^{t}\)

Compute \(A A^{t}\) using matrix multiplication:\[A A^{t} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} a & c \ b & d \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & ac + bd \ ac + bd & c^2 + d^2 \end{pmatrix} \]

Step 6: Verify Symmetry of the Resultant Matrix

The resulting matrix is \(\begin{pmatrix} a^2 + b^2 & ac + bd \ ac + bd & c^2 + d^2 \end{pmatrix}\). Since the off-diagonal elements \(ac + bd\) are equal, the matrix is indeed symmetric.

Key Concepts

Matrix Transpose

A matrix transpose is a fundamental concept in linear algebra. It involves transforming the matrix by swapping its rows and columns. Imagine you have a box with buttons aligned in rows and columns and you swap each row with a column. That's exactly what happens in a transpose.
Let's denote a simple matrix \( A \) as follows: A \( = \begin{pmatrix} x & y \ z & w \end{pmatrix} \).

• In the transposed matrix, each element switches its position from row to column, and vice versa.
• Therefore, its transpose \( A^{t} \) becomes \( \begin{pmatrix} x & z \ y & w \end{pmatrix} \).

Swapping sounds trivial, but when applying transposes in various operations like calculating products, it becomes crucial.To check if a matrix is symmetric, a matrix is compared to its transpose. If they are the same, then the matrix is called symmetric.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce another matrix. It's more than simple element-wise multiplication. Instead, you multiply elements and sum them to get each cell of the resulting matrix.
Let's look at two matrices:

• \( B = \begin{pmatrix} p & q \ r & s \end{pmatrix} \)
• \( C = \begin{pmatrix} u & v \ w & x \end{pmatrix} \)

To multiply \( B \) and \( C \), you follow these steps for each position in the resulting matrix:

• Multiply row elements of B by column elements of C and sum them up for each cell in the new matrix.
• The (1,1) position in the resulting matrix is \( pu + qw \), the (1,2) is \( pv + qx \), and so on until a full matrix is formed.

Understanding this helps us easily perform multiplications and verify symmetric properties in the resulting matrix.

Square Matrices

A square matrix is one where the number of rows is equal to the number of columns. Common examples include identity matrices and symmetric matrices.
This format is critical for certain operations:

• Square matrices are easy to visualize as perfect squares if you imagine rows and columns aligning precisely.
• They possess unique properties such as determinant calculation and eigenvalues.

In linear algebra, fiddling with square matrices becomes vital because certain algorithms rely heavily on their structural properties. For instance, confirming whether A\(A^{t}\) (which we've checked previously) is symmetric, focuses directly on these structural characteristics of square matrices.

2x2 Matrices

The \(2\times2\) matrices play a special role as they are simple yet powerful tools for illustrating matrix concepts. They often serve as introductory examples due to their manageability.
Consider a general \(2\times2\) matrix:

• This matrix has four elements, and matrix operations, such as addition, transpose, and multiplication, remain easier with them.
• For example: \( D = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) allows an easy showcase of detecting symmetry when \( D = D^{t} \).

These matrices are not only used in educational contexts but appear frequently in applications involving systems of equations and geometric transformations. The homework problem utilizing a \(2\times2\) matrix helps to concretely firm understanding in students' minds.