Question

Find the trace of the given matrix. \(\left[\begin{array}{cc}7 & 5 \\ -5 & -4\end{array}\right]\)

Short answer

The trace is 3.

Step-by-step solution

Understanding the Trace

The trace of a matrix is the sum of the elements on the main diagonal. For a 2x2 matrix, it is the sum of the element in the first row first column and the element in the second row second column.

Identify the Diagonal Elements

In the provided matrix \[\left[\begin{array}{cc}7 & 5 \ -5 & -4\end{array}\right]\]the diagonal elements are 7 and -4.

Calculate the Sum of the Diagonal Elements

Add the diagonal elements: \(7 + (-4) = 7 - 4 = 3\).

Key concepts

Matrix Algebra

Matrix algebra is an essential part of mathematics that involves operations on matrices. It is a powerful tool used in various fields like computer science, engineering, and physics. In matrix algebra, you can perform several types of operations such as addition, subtraction, and multiplication. An operation on matrices often follows specific rules.

• **Matrix Addition and Subtraction:** Only matrices of the same size can be added or subtracted, element by element.
• **Matrix Multiplication:** This is a bit more complex. The number of columns in the first matrix must match the number of rows in the second matrix.
• **Determinants and Inverses:** These are special values associated with square matrices and have numerous applications.

Understanding these basic operations in matrix algebra allows one to delve deeper into more complex topics, like solving systems of linear equations or performing transformations in graphics.

Main Diagonal

The main diagonal of a matrix is an important concept in matrix algebra. It consists of the elements that extend diagonally from the top left corner to the bottom right.
To identify the main diagonal, follow these simple steps:

• Start from the element in the first row, first column. This is the starting point of the main diagonal.
• Move diagonally downwards, to the right, selecting one element from each subsequent row and column.

These diagonal elements have a special significance in many matrix operations. For example:

• In the operation used to find the trace, only the main diagonal elements are involved.
• Diagonalization of matrices, an advanced topic, relies heavily on these diagonal elements.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices containing just two rows and two columns. Being so straightforward, they offer an excellent starting point for understanding matrix operations. Generally represented as:
\[\begin{bmatrix}a & b \c & d \end{bmatrix}\]Here, each letter denotes a specific element in the matrix.
Let's break down its parts:

• **First Row (a, b):** Elements are identified by their position: the first element ("a") is the first row, first column; and "b" is the first row, second column.
• **Second Row (c, d):** Similarly, "c" is in the second row, first column, and "d" is in the second row, second column.

Such matrices are simple to work with, especially when calculating the trace, as you only have to sum the elements from the first row, first column, and the second row, second column. This simplicity is why they are often used as introductory examples in matrix algebra.