Question

Find the trace of the given matrix. \(\left[\begin{array}{cc}-3 & -10 \\ -6 & 4\end{array}\right]\)

Short answer

The trace of the matrix is 1.

Step-by-step solution

Understand the Matrix

The trace of a matrix is the sum of the elements on its main diagonal. The given matrix is: \[\begin{array}{cc}-3 & -10 \-6 & 4\end{array}\] Identify the main diagonal elements, which are -3 and 4.

Calculate the Trace

Add the elements on the main diagonal of the matrix. In this case, add -3 and 4:\[\text{Trace} = -3 + 4\]

Simplify the Sum

Simplify the sum from the previous step to find the trace:\[-3 + 4 = 1\]

Key concepts

Matrix Algebra

Matrix algebra is an area of mathematics that deals with matrices and their operations. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are fundamental in various fields like science, engineering, and computer graphics. In matrix algebra, you can perform operations such as addition, subtraction, and multiplication of matrices, as well as more complex operations like finding determinants and inverses.
Understanding the basics of matrix algebra is essential, as it provides the framework for more advanced mathematical and real-world problem-solving. Key things to remember about matrices include:

• They are defined by their size, described as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.
• They must be of compatible sizes to perform certain operations, like addition and multiplication.
• Special types of matrices, such as square matrices where the number of rows is equal to the number of columns, have unique properties.

Understanding matrix algebra will enhance your ability to compute, transform and analyze numerical data effectively.

Diagonal Elements

Within any type of matrix, diagonal elements refer to those that stretch from the top left to the bottom right corner in a square matrix. These are crucial in many matrix operations, including finding the trace or determining if the matrix has any special properties like being diagonal or identity.
A diagonal element of a square matrix is represented as an entry at position \( (i, i) \). This means its row index and column index are the same. In our example matrix:

• The diagonal elements are \(-3\) and \(4\).
• They are located at positions \((1, 1)\) and \((2, 2)\) respectively.

Diagonal elements play a crucial role as they often determine the key properties of the matrix, and they simplify many types of calculations.

Matrix Calculation

Matrix calculation involves performing various mathematical operations on matrices, including calculating their trace, product, or inverse. Specifically, when calculating the trace of a matrix, you sum up all the diagonal elements in a square matrix.
This calculation is straightforward but very important in different applications, such as simplifying expressions and providing insights into the matrix's characteristics. Here’s a step-by-step guide for calculating the trace:

• Identify the main diagonal elements of the matrix (those from the top left to the bottom right).
• Add these diagonal elements together.
• The resulting sum is the trace of the matrix.

In this exercise, for the matrix \[\left[\begin{array}{cc}-3 & -10 \-6 & 4\end{array}\right]\]we have observed that the diagonal elements \(-3\) and \(4\) sum up to \(1\) to get the trace. Calculating the trace is an essential skill, as it can simplify matrix analysis and open doors to further understanding matrix behavior and applications.