Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrrr} -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 48

Step 1: Identify type of the matrix

Observe the matrix and identify its type. The given matrix has all non-diagonal elements as 0, thus it is a diagonal matrix.

Step 2: Evaluate the determinant

For a diagonal matrix, the determinant is just the product of the diagonal elements. So, calculate the product of the diagonal entries -2, 3, -1, 2, and -4.

Key Concepts

Diagonal Matrix

A diagonal matrix is a specific type of matrix which stands out due to its simplicity. In this matrix, all the elements outside the main diagonal are zero. This means, when you look at it, most of the elements that aren't in a straight line from the top left to the bottom right are zero. For example, in our matrix, you can observe that only the diagonal positions are filled with numbers, which are

• -2
• 3
• -1
• 2
• -4

These are the diagonal entries. Diagonal matrices are especially useful in linear algebra due to their simplicity.
They simplify many operations, such as calculating determinants, making them much easier to handle in mathematical problems.

Matrix Determinant Evaluation

The determinant of a matrix provides crucial information about the matrix, such as whether it is invertible or singular. Each type of matrix has its own way of evaluating the determinant, but the concept remains consistent. In general, the determinant reflects the scale factor by which the matrix transforms space. For a 5x5 matrix like the example given, calculating the determinant through expansion by minors or row reduction would be complex and time-consuming.
However, diagonal matrices provide a shortcut. Here, you can evaluate the determinant simply by multiplying the diagonal entries. This is because the multiplicative property of determinants simplifies in a diagonal format, drastically reducing computational efforts.

Product of Diagonal Entries

When dealing with diagonal matrices, the process of determining the matrix determinant becomes straightforward—it's just the product of the diagonal entries. For the given matrix, we find each of the diagonal numbers,

• -2
• 3
• -1
• 2
• -4

By multiplying these, we get the determinant of the matrix.
Calculate the product: \(-2 \times 3 \times -1 \times 2 \times -4 = -48\).
This calculation shows how efficient it is to handle the determinant of diagonal matrices. By focusing on just the main diagonal, a potentially complex task becomes quite manageable and quick.