Problem 5 - Differential Equations and Linear Algebra

Question

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$A B \text { and } \operatorname{tr}(A B).$$

Step-by-Step Solution

Answer

The product of matrices A and B is \(\left[\begin{array}{rr}16 & 8 \\\ 6 & -17\end{array}\right]\), and the trace of this matrix is \(-1\).

Step 1: Checking the possibility of matrix multiplication

To compute the product of matrices A and B, the number of columns in matrix A must be equal to the number of rows in matrix B. Matrix A has 4 columns and matrix B has 4 rows. Therefore, matrix multiplication between A and B is possible.

Step 2: Performing matrix multiplication

To find the product AB, we need to compute the dot product of each row of matrix A with each column of matrix B. We will have: \(AB = \left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\\ -1 & -1 & 5 & 0\end{array}\right] \left[\begin{array}{rr}-3 & 0 \\\ 2 & 2 \\\ 1 & -3 \\\ 0 & 1\end{array}\right]\) = \(\left[\begin{array}{rr} (-2)(-3)+(4)(2)+(2)(1)+(6)(0) & (-2)(0)+(4)(2)+(2)(-3)+(6)(1) \\\ (-1)(-3)+(-1)(2)+(5)(1)+(0)(0) & (-1)(0)+(-1)(2)+(5)(-3)+(0)(1) \end{array}\right]\) = \(\left[\begin{array}{rr} 6+8+2+0 & 0+8-6+6 \\\ 3-2+5+0 & 0-2-15+0 \end{array}\right]\) = \(\left[\begin{array}{rr} 16 & 8 \\\ 6 & -17 \end{array}\right]\)

Step 3: Finding the trace of the product

The trace of a square matrix is the sum of the diagonal elements. In this case, the trace of AB is: \(\operatorname{tr}(AB) = 16 + (-17)\) \(\operatorname{tr}(AB) = -1\) So the expression \(AB\) is possible, and its value is \(\left[\begin{array}{rr}16 & 8 \\\ 6 & -17\end{array}\right]\), with a trace of \(-1\).

Key Concepts

Trace of a Matrix

The trace of a matrix is a fundamental concept in linear algebra. It is defined for a square matrix and is the sum of the elements on its main diagonal. The main diagonal of a square matrix consists of elements that run from the top left to the bottom right.
For example, if you have a matrix:

• \[\begin{bmatrix}x_{11} & x_{12} & x_{13} \x_{21} & x_{22} & x_{23} \x_{31} & x_{32} & x_{33} \\end{bmatrix}\]

The trace is found by adding up these diagonal elements: \( x_{11} + x_{22} + x_{33} \). In the context of our problem, the trace of AB, which is a 2x2 matrix, involves adding the diagonal values: \(16 + (-17) = -1\).
Calculating the trace is a straightforward process but plays a crucial role in simplifying complex mathematical problems. It is widely used in various applications, such as quantum mechanics and system theory.

Matrix Dimensions

Understanding matrix dimensions is essential for performing operations like multiplication. Dimensions are expressed by the number of rows and columns a matrix has, typically noted as \( m \times n \).
For example, a matrix with 3 rows and 4 columns is noted as \( 3 \times 4 \). In our case:

• Matrix \( A \) has dimensions \( 2 \times 4 \), meaning it has 2 rows and 4 columns.
• Matrix \( B \) has dimensions \( 4 \times 2 \), with 4 rows and 2 columns.

For two matrices to be multiplied, the number of columns in the first must equal the number of rows in the second. This condition is met in our example, allowing the multiplication of matrices A and B. The result is a new matrix AB with dimensions determined by the outer dimensions, which would be \( 2 \times 2 \) in this case.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. When applied within the context of matrix multiplication, it involves computing the sum of the products of the corresponding entries of the sequences of numbers.
Each entry in the resulting matrix from multiplying matrices A and B is calculated through the dot product of a row from matrix A and a column from matrix B. For instance, using our matrices:

• The entry in the first row and first column of AB is the dot product of the first row of A and the first column of B:\[(-2)(-3) + (4)(2) + (2)(1) + (6)(0) = 16\]
• This operation is repeated for each entry of the resulting matrix AB.

The dot product helps in determining each element of the product matrix, which is crucial for accurate computation in matrix multiplication.

Square Matrix

A square matrix is one where the number of rows and columns is equal, such as a 2x2, 3x3, etc. This characteristic is important, particularly when calculating the trace or certain other properties like determinants and eigenvalues.
In the problem presented, the product matrix \( AB \) is a square matrix since it has dimensions \( 2 \times 2 \). This enables us to calculate its trace and use other operations that are defined only for square matrices.
Square matrices often represent more complex systems and are pivotal in advanced mathematical theories and applications. Their properties allow calculations like the trace, playing a central role in simplifying and solving mathematical challenges.