Problem 40 - Precalculus: Mathematics for Calculus

Question

Find the determinant of a \(10 \times 10\) matrix which has a 2 in each main diagonal entry and zeros everywhere else.

Step-by-Step Solution

Answer

The determinant is 1024.

Step 1: Understanding the Matrix

The given matrix is a diagonal matrix of size \(10 \times 10\) where each element on the main diagonal is 2, and all other elements are 0. This matrix can be represented as \( A = \text{diag}(2, 2, \ldots, 2) \).

Step 2: Property of a Diagonal Matrix

A key property of diagonal matrices is that their determinant is equal to the product of the elements on the main diagonal. Thus, for a \(10 \times 10\) diagonal matrix, the determinant is the multiplication of the 10 diagonal elements.

Step 3: Calculating the Determinant

Since all the diagonal elements are 2, the determinant can be calculated as: \[\det(A) = 2 \times 2 \times \cdots \times 2 = 2^{10}\]

Step 4: Result of Calculation

Calculate \(2^{10}\) to find the determinant:\[2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024\]

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where each element outside the main diagonal is zero. The main diagonal runs from the top left to the bottom right of the matrix. Here are a few characteristics of diagonal matrices:

• All diagonal matrices have elements only on the main diagonal; all off-diagonal elements are zero.
• The main diagonal of a diagonal matrix contains all its non-zero elements.
• Given a matrix with dimensions \(n \times n\), a diagonal matrix will have \(n\) non-zero elements if all the diagonal positions are filled.

In this exercise, the matrix is described as having 2s on its diagonal, making it a constant diagonal matrix. This means each diagonal entry is equal to the constant value of 2. Diagonal matrices are particularly easy to work with because multiplication and determination of determinants are simplified.

Matrix Multiplication

Matrix multiplication involves the calculation of the rows of the first matrix with the columns of the second matrix. However, when working with diagonal matrices, the process becomes much simpler.
Some key facts include:

• When multiplying two diagonal matrices, the result is another diagonal matrix.
• The diagonal elements of the product matrix are the products of the corresponding diagonal elements of the original matrices.
• For example, if you multiply two diagonal matrices \(Diag(a_1, a_2, \ldots, a_n)\) and \(Diag(b_1, b_2, \ldots, b_n)\), the resulting matrix is \(Diag(a_1b_1, a_2b_2, \ldots, a_nb_n)\).

In the context of the given problem, matrix multiplication isn't needed directly, but the simplicity of diagonal matrices in multiplication highlights why they are so easy to handle.

Properties of Matrices

Matrices have several properties that are useful in matrix operations such as determinant calculation. Here are some general properties of matrices:

• Determinants: Only square matrices (same number of rows and columns) have a determinant. The determinant is a scalar value that summarizes certain properties of the matrix.
• Additive and Scalar Properties: Matrix operations can include scaling (multiplying each entry by a number) and addition, following specific rules such as distributive and associative laws.
• Transpose: Changing the matrix's rows into columns is called the matrix transpose, often denoted as \(A^T\). The determinant of a matrix and its transpose are the same.
• Inverse: Some matrices have an inverse, denoted \(A^{-1}\). If a matrix has an inverse, then the product of the matrix and its inverse yields the identity matrix.

This exercise focuses on calculating the determinant, and diagonal matrices exhibit a straightforward property where their determinant is simply the product of their diagonal elements. This makes calculations like \(2^{10}\) quite efficient to perform.