Problem 101 - Precalculus with Limits

Question

In Exercises 101-104, evaluate the determinant. \(\left| \begin{array}{r} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant is 10.

Step 1: Identify the Type of Matrix

Examine the matrix, note that all the non-diagonal elements of the matrix are zeros. Thus, it is a diagonal matrix.

Step 2: Calculate the Determinant

For a diagonal matrix, the determinant can be calculated as the product of the diagonal elements. Hence, in this case, it will be the product of 1, 5, and 2.

Step 3: Evaluate the Product

Calculate the product of the diagonal elements: \(1 \times 5 \times 2\)

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements outside of the main diagonal are zero. This makes them quite straightforward and enjoyable to work with.

• The main diagonal stretches from the top left to the bottom right of the matrix.
• Every element not on the main diagonal should be zero.

Diagonal matrices have unique properties that simplify many matrix calculations.
For instance, when you want to find the determinant of a diagonal matrix, the calculation is quick and simple.

Matrix Multiplication

Matrix multiplication involves multiplying two matrices to produce another matrix. While diagonal matrices make excellent simplifiers in multiplication, here's how matrix multiplication works in general:

• Think of it as the row-by-column method.
• Multiply elements of rows in the first matrix by corresponding elements of columns in the second matrix.


• Sum these products for each element of the resulting matrix.

An easy special case occurs when diagonal matrices are involved. Since all non-diagonal elements are zero, computations get simpler.

Evaluating Determinants

Determining the determinant is an essential process in linear algebra, used to understand attributes such as matrix invertibility.
For generic square matrices, calculating a determinant involves a combination of elements through designated methods.

• In a diagonal matrix, however, this becomes much easier.


• One simply multiplies the elements on the main diagonal together to find the determinant.
• For example, for the diagonal matrix given, the determinant is calculated as: \(1 \times 5 \times 2\).

This simplicity in computing determinants is one of the key benefits of diagonal matrices.