Problem 49 - Algebra and Trigonometry

Question

Evaluate the determinants. $$ \left|\begin{array}{ccccc}{a} & {0} & {0} & {0} & {0} \\ {0} & {b} & {0} & {0} & {0} \\ {0} & {0} & {c} & {0} & {0} \\ {0} & {0} & {0} & {d} & {0} \\\ {0} & {0} & {0} & {0} & {e}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \( a \cdot b \cdot c \cdot d \cdot e \).

Step 1: Recognize Matrix Type

The given matrix is a diagonal matrix, where all non-diagonal elements are zeros. This type of matrix has specific properties that simplify calculating the determinant.

Step 2: Use Diagonal Matrix Property

For a diagonal matrix, the determinant is the product of its diagonal elements. Therefore, the determinant of a diagonal matrix \( \begin{pmatrix} a & 0 & 0 & 0 & 0 \ 0 & b & 0 & 0 & 0 \ 0 & 0 & c & 0 & 0 \ 0 & 0 & 0 & d & 0 \ 0 & 0 & 0 & 0 & e \end{pmatrix} \) is given by the product \( a \cdot b \cdot c \cdot d \cdot e \).

Key Concepts

Diagonal Matrix

A diagonal matrix is a type of square matrix characterized by having all its non-diagonal elements set to zero. In other words, \( A = [a_{ij}] \) is a diagonal matrix if \( a_{ij} = 0 \) for all \( i eq j \). This means that only the elements on the leading diagonal (from the top left to the bottom right) are potentially non-zero.
For example, in a \( 5 \times 5 \) diagonal matrix, like the one shown, only the elements \( a, b, c, d, \) and \( e \) on the diagonal aren't zero.

Diagonal matrices are quite significant in linear algebra due to their computational simplicity. They are easy to multiply and transpose, and they have a direct method for determinant calculation, which we'll explore next.

Matrix Determinant Calculation

Calculating the determinant of a matrix involves more steps with typical square matrices, but diagonal matrices simplify this process immensely. The determinant of a diagonal matrix is straightforward—it is simply the product of the elements on its main diagonal.
For example, given a diagonal matrix\[\begin{pmatrix}a & 0 & 0 & 0 & 0 \0 & b & 0 & 0 & 0 \0 & 0 & c & 0 & 0 \0 & 0 & 0 & d & 0 \0 & 0 & 0 & 0 & e\end{pmatrix}\]
the determinant can be calculated as:
\[ ext{Det}(A) = a \cdot b \cdot c \cdot d \cdot e\]
This property makes diagonal matrices highly efficient in applications requiring determinant calculations, such as solving systems of linear equations, finding eigenvalues, and analyzing matrix invertibility.

Properties of Determinants

Determinants are a critical concept in linear algebra, offering insight into multiple matrix properties such as invertibility, linear transformations, and more. Here are some essential properties of determinants:

• The determinant of a matrix provides a scalar value that can describe volume scaling factors in transformations. Specifically, it tells how a matrix would scale area or volume when applied as a transformation.
• For a basic property, if a matrix is non-invertible (singular), its determinant is zero. This indicates the transformation squashes some dimension to zero.
• The determinant changes sign if two rows (or columns) of the matrix are swapped. Such a property is crucial when performing row operations in achieving row echelon forms.
• Multiplying a row (or column) by a scalar multiplies the determinant by that scalar. For diagonal matrices, this effect is simplified by the multiplication property we've seen, as each element contributes straightforwardly to the scale factor.

Understanding these properties can make it simpler to predict or check computations in problems involving more complex matrices.