Question

Find the trace of the given matrix. \(\left[\begin{array}{cc}-6 & 0 \\ -10 & 9\end{array}\right]\)

Short answer

The trace of the matrix is 3.

Step-by-step solution

Understand the Concept of Trace

The trace of a matrix is the sum of the elements on its main diagonal. For a 2x2 matrix of the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the trace is \( a + d \).

Identify the Main Diagonal Elements

For the given matrix \( \begin{bmatrix} -6 & 0 \ -10 & 9 \end{bmatrix} \), the elements on the main diagonal are \(-6\) and \(9\).

Calculate the Trace

Add the main diagonal elements together: \(-6 + 9\).

Perform the Addition

Calculate \(-6 + 9 = 3\).

Key concepts

Diagonal Elements

When we talk about diagonal elements in a matrix, we are focusing on those numbers located from the top left to the bottom right in a square matrix. These elements form what is known as the 'main diagonal.'

For any matrix, the main diagonal is crucial because it often has special properties and applications. In particular, when dealing with square matrices (matrices with the same number of rows and columns), it's a lot more straightforward as these notes line up right from one corner to the opposite.

Understanding which elements are diagonal elements is key in computing concepts like the trace of a matrix. For example, consider the matrix: \[\begin{bmatrix}-6 & 0 \-10 & 9\end{bmatrix}\]The diagonal elements are \(-6\) and \(9\) because they are positioned from the top left to the bottom right. It's their sum that gives us the trace. Identifying the main diagonal correctly is essential in solving matrix-related problems quickly and accurately.

2x2 Matrix

A 2x2 matrix is one of the simplest square matrices you can deal with. It consists of two rows and two columns. Although they are small, these matrices are very useful in explaining fundamental matrix operations like trace, determinant, and eigenvalues.

The general form of a 2x2 matrix is:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]where \(a\), \(b\), \(c\), and \(d\) are elements that belong to the matrix. Often used in basic matrix calculations, 2x2 matrices serve as building blocks for understanding larger matrix operations.

This size of matrix is also common in introductory linear algebra because they are easy to handle and visualize. When learning about matrix properties, starting with 2x2 matrices can make it easier to grasp fundamental concepts like trace, multiplication, and inverses.

Matrix Addition

Matrix Addition is a basic operation in linear algebra where two matrices of the same size are added by adding corresponding entries. To visualize this, consider two matrices:\[A = \begin{bmatrix}1 & 2 \3 & 4\end{bmatrix},B = \begin{bmatrix}-1 & 0 \0 & 1\end{bmatrix}\]The sum of these two matrices, \(A + B\), is calculated by adding the corresponding elements:\[A + B = \begin{bmatrix}1 + (-1) & 2 + 0 \3 + 0 & 4 + 1\end{bmatrix} = \begin{bmatrix}0 & 2 \3 & 5\end{bmatrix}\]

Matrix addition, like what we saw with diagonal elements in finding a trace, involves scrutinizing elements in matched positions, demanding the matrices be of the same size. This operation is simple yet vital. It helps consolidate understanding on how matrices function together, cementing the foundation needed for more complex operations in higher-level math contexts.