Problem 29 - Algebra 2

Question

What is the determinant of \(\left[\begin{array}{ccc}{5} & {0} & {0} \\ {0} & {5} & {0} \\ {0} & {0} & {5}\end{array}\right] ?\) \(\begin{array}{llll}{\text { A. } 5} & {\text { B. } 25} & {\text { C. } 125} & {\text { D. } 555}\end{array}\)

Step-by-Step Solution

Answer

The determinant of the matrix is 125.

Step 1: Understanding the Matrix

We are given a 3x3 diagonal matrix where all the elements except the diagonal are zero. The diagonal elements are all 5.

Step 2: Recall the Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is the product of the diagonal elements.

Step 3: Calculate the Determinant

Multiply the diagonal elements: 5 * 5 * 5 = 125.

Key Concepts

Diagonal Matrix

Understanding diagonal matrices is fundamental in linear algebra. A diagonal matrix is a special type of square matrix where all entries off the main diagonal are zero. This means that if you look at a matrix and see that all its components except those from the top-left to the bottom-right are zeroes, you've got a diagonal matrix on your hands.

Why are they important? Well, diagonal matrices simplify many matrix operations. For example, when you're working with very large matrices, calculations can get complicated fast. But if your matrix is diagonal, many operations, such as finding powers of matrices, become much simpler because you only need to focus on the diagonal elements.

Here's another reason: diagonal matrices serve as a sort of 'scaling' operator in transformations. Each non-zero element in the diagonal can be thought of as scaling the corresponding vector component. This is particularly useful in various applications of mathematics and computer graphics for scaling objects.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. In a mathematical sense, it gives you information about the volume scaling factor by the matrix transformation, and it tells you whether the matrix is invertible or not (if the determinant is zero, the matrix cannot be inverted).

Calculating the determinant can be simple or complex, depending on the matrix. For a 2x2 matrix, it's a quick calculation, but for larger matrices, it requires more steps. One interesting aspect of determinants is that if any column or row of a matrix is filled with zeros, the determinant of that matrix is zero. In the context of our exercise, since we have a diagonal matrix, finding the determinant is as easy as multiplying its diagonal entries together.

Algebraic Matrices

Algebraic matrices is a term that generally covers the study of matrices within the field of algebra. Matrices are arrays of numbers or functions that can represent systems of linear equations, or linear transformations in geometry. Algebraic matrices are used in various ways, such as solving linear systems, in graph theory, and even in statistics where they appear as covariance matrices.

Understanding the properties of matrices, like commutative and associative laws, which do not always hold as they do in regular arithmetic, is essential for students. Also, understanding the different types of matrices, such as row and column matrices, identity matrices, symmetric matrices, etc., forms a base for further studies in mathematics and related fields. Exercises like the one we're discussing provide not only practice in computation but also deepen the understanding of the very nature of linear transformations.