Problem 64 - College Algebra

Question

Evaluate: $$\left|\begin{array}{lllll}2 & 0 & 0 & 0 & 0 \\\0 & 3 & 0 & 0 & 0 \\\0 & 0 & 2 & 0 & 0 \\\0 & 0 & 0 & 1 & 0 \\\0 & 0 & 0 & 0 & 4\end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is 48.

Step 1: Identify the type of Matrix

The given matrix is a diagonal matrix. The determinant of a diagonal matrix is the product of the elements on the main diagonal.

Step 2: Calculate the determinant

To calculate the determinant of the matrix, simply multiply the entries along the main diagonal. This gives: \(2 \times 3 \times 2 \times 1 \times 4 = 48 \)

Step 3: State The Result

The determinant of the matrix is 48.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where the entries outside the main diagonal are all zero. The main diagonal is the set of entries that extend from the top left corner to the bottom right corner of the matrix. It might seem like a simple structure, but diagonal matrices play a significant role in linear algebra because they make certain operations much easier.

In the context of our exercise, the matrix presented is a 5x5 diagonal matrix, with non-zero entries only along the diagonal. Since diagonal matrices only have non-zero elements in those diagonal positions, they are often used to represent scaling transformations in geometry, wherein each axis is scaled by the corresponding diagonal element.

Matrix Determinant Calculation

The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible and the volume scale factor of the geometric transformation represented by the matrix.

To calculate the determinant of a general square matrix, many methods can be applied such as expansion by minors or the Laplace expansion. However, in the case of a diagonal matrix, the calculation simplifies dramatically. For a diagonal matrix, the determinant is equal to the product of the diagonal entries. This property not only makes the calculation faster but also helps in understanding how changes in the diagonal elements affect the overall determinant. In our exercise, by multiplying the diagonal entries, we got the determinant of 48.

Properties of Determinants

Determinants have a rich set of properties that make them valuable tools in linear algebra. One of the essential properties is that the determinant of a diagonal matrix is the product of its main diagonal elements, as seen in the given exercise. This is a special case of a more general property, where the determinant of a triangular matrix (either upper or lower triangular) also equals the product of its diagonal elements.

Some other key properties of determinants include:

• The determinant of the identity matrix is always 1.
• If two rows or columns of a matrix are equal or proportional, its determinant is 0, indicating the matrix is not invertible.
• Interchanging any two rows or columns multiplies the determinant by -1.
• The determinant of the product of matrices is equal to the product of their determinants (i.e., det(AB) = det(A)det(B)).

Understanding these properties helps in solving complex determinant calculations more efficiently and also aids in grasping the relationship between a matrix and its determinant.