Problem 35 - Advanced Engineering Mathematics

Question

$$ \begin{aligned} &\text { Suppose } \mathbf{A}=\left(\begin{array}{ll} 2 & 1 \\ 6 & 3 \\ 2 & 5 \end{array}\right) \text { . Verify that the matrix } \mathbf{B}=\mathbf{A} \mathbf{A}^{T} \text { is }\\\ &\text { symmetric. } \end{aligned} $$

Step-by-Step Solution

Answer

Matrix B is symmetric since B = B^T.

Step 1: Find the transpose of matrix A

The transpose of a matrix is found by swapping rows with columns. For matrix \( \mathbf{A} = \begin{pmatrix} 2 & 1 \ 6 & 3 \ 2 & 5 \end{pmatrix} \), its transpose, \( \mathbf{A}^T \), is: \( \begin{pmatrix} 2 & 6 & 2 \ 1 & 3 & 5 \end{pmatrix} \).

Step 2: Compute matrix B as A times A transpose

Matrix \( \mathbf{B} \) is given by \( \mathbf{A} \mathbf{A}^T \). To calculate this, multiply \( \mathbf{A} \) by \( \mathbf{A}^T \). Use the matrix multiplication formula: \( (AB)_{ij} = \sum_{k} A_{ik}B_{kj} \). Calculating step-by-step:- First row, first column: \((2*2) + (1*1) = 4 + 1 = 5\)- First row, second column: \((2*6) + (1*3) = 12 + 3 = 15\)- First row, third column: \((2*2) + (1*5) = 4 + 5 = 9\)- Second row, first column: \((6*2) + (3*1) = 12 + 3 = 15\)- Second row, second column: \((6*6) + (3*3) = 36 + 9 = 45\)- Second row, third column: \((6*2) + (3*5) = 12 + 15 = 27\)- Third row, first column: \((2*2) + (5*1) = 4 + 5 = 9\)- Third row, second column: \((2*6) + (5*3) = 12 + 15 = 27\)- Third row, third column: \((2*2) + (5*5) = 4 + 25 = 29\).

Step 3: Write the resulting matrix B

After performing the matrix multiplication, \( \mathbf{B} = \begin{pmatrix} 5 & 15 & 9 \ 15 & 45 & 27 \ 9 & 27 & 29 \end{pmatrix} \). This is the product of \( \mathbf{A} \) and \( \mathbf{A}^T \).

Step 4: Check if matrix B is symmetric

A symmetric matrix is one where \( B = B^T \). Compare \( \mathbf{B} \) with its transpose:\[ \mathbf{B}^T = \begin{pmatrix} 5 & 15 & 9 \ 15 & 45 & 27 \ 9 & 27 & 29 \end{pmatrix} \. \]Matrix \( \mathbf{B} \) is the same as its transpose, confirming that it is symmetric.

Key Concepts

Matrix Transpose

The transpose of a matrix is a fundamental concept in linear algebra. It involves flipping the matrix over its diagonal. This means that the row and column indices of each element are swapped. For a given matrix \( \mathbf{A} \), its transpose is denoted as \( \mathbf{A}^T \). For example, if matrix \( \mathbf{A} \) is

• \( \begin{pmatrix} 2 & 1 \ 6 & 3 \ 2 & 5 \end{pmatrix} \)

then its transpose \( \mathbf{A}^T \) becomes

• \( \begin{pmatrix} 2 & 6 & 2 \ 1 & 3 & 5 \end{pmatrix} \).

Transposing changes the orientation of the matrix while maintaining the arrangement of numbers. This operation is crucial when dealing with matrices because many operations, such as finding symmetric matrices, rely on the transpose. It’s a skill that comes in handy frequently, so practicing with different matrices will help you become comfortable with it.

Matrix Multiplication

Matrix multiplication is an operation that combines two matrices to form a new matrix. However, it’s a bit more complex than simply multiplying individual numbers together. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. For example, if \( \mathbf{A} \) is a 3x2 matrix and \( \mathbf{A}^T \) is a 2x3 matrix, the product \( \mathbf{A} \mathbf{A}^T \) results in a 3x3 matrix.

• First, we take each element of a row from the first matrix \( \mathbf{A} \) and multiply it by the corresponding element of a column from the second matrix \( \mathbf{A}^T \).
• Sum these products up to get the corresponding element in the resulting matrix \( \mathbf{B} \).

This operation is repeated for each row-column pair. Matrix multiplication results can reveal insightful properties about matrices and is used extensively in fields like computer graphics, physics simulations, and more.

Symmetric Matrix

A symmetric matrix is one of the most intriguing types in linear algebra. Simply put, a matrix is symmetric if it is equal to its transpose. In other words, matrix \( \mathbf{B} \) is symmetric if \( \mathbf{B} = \mathbf{B}^T \). For instance, when you take a look at the matrix \( \mathbf{B} \) calculated from the multiplication of matrix \( \mathbf{A} \) and its transpose \( \mathbf{A}^T \) as follows:

• \( \begin{pmatrix} 5 & 15 & 9 \ 15 & 45 & 27 \ 9 & 27 & 29 \end{pmatrix} \)

You’ll find that switching rows and columns doesn’t change it; it remains:

• \( \begin{pmatrix} 5 & 15 & 9 \ 15 & 45 & 27 \ 9 & 27 & 29 \end{pmatrix} \).

This symmetry is particularly valuable in mathematics and physics because symmetric matrices often represent real-world phenomena such as energy or distance, where two points are equally influential to each other in both directions.