Problem 62 - Precalculus: Mathematics for Calculus

Question

Evaluate the determinants. $$\left|\begin{array}{lllll} a & a & a & a & a \\ 0 & a & a & a & a \\ 0 & 0 & a & a & a \\ 0 & 0 & 0 & a & a \\ 0 & 0 & 0 & 0 & a \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is \( a^5 \).

Step 1: Identify Matrix Structure

Examine the given 5x5 matrix. Observe that it is an upper triangular matrix where all elements below the diagonal are zeros.

Step 2: Recall Property of Triangular Matrix

Remember that the determinant of a triangular matrix (whether upper or lower triangular) is the product of the diagonal elements.

Step 3: Multiply Diagonal Elements

Identify the diagonal elements of the matrix: all of them are 'a'. Thus, the product of the diagonal elements is \( a \times a \times a \times a \times a = a^5 \).

Step 4: Evaluate Determinant

From the previous step, the determinant of the matrix is calculated to be \( a^5 \).

Key Concepts

Understanding an Upper Triangular Matrix

An upper triangular matrix is a special kind of square matrix. In this matrix type, all the elements below the main diagonal are zero. Here's what you need to know:

• Only the diagonal and the elements above can be non-zero.
• The matrix can be any size, as long as it's square, meaning it has the same number of rows as columns.

The main diagonal is the line of entries that runs from the upper left to the lower right of the matrix. So, in an upper triangular matrix, if you were to start at any element below this diagonal, you would encounter zeros. For example, in the matrix given in the exercise, the first diagonal line includes the elements: \( a, a, a, a, a \), and anything below this line is zero. This structure makes it easier to calculate determinants.

Triangular Matrix Types

Triangular matrices are very useful in linear algebra. They save time with calculations and often make complex problems simpler. There are two main kinds of triangular matrices:

• Upper Triangular Matrix – As mentioned earlier, this is when all elements below the main diagonal are zero.
• Lower Triangular Matrix – Here, the opposite is true; all elements above the diagonal are zero.

These categories help because each comes with helpful mathematical shortcuts, like quickly calculating determinants. Understanding whether a matrix is upper or lower triangular can streamline many processes in algebra. When working with triangular matrices, it's crucial to remember these characteristics, as they directly affect their properties and how calculations are carried out.

Core Properties of Determinants in Triangular Matrices

Determinants carry significant importance in mathematics, particularly when applied to triangular matrices. Here are the key properties of determinants in such matrices:

• Simple Calculation: For a triangular matrix, whether upper or lower, the determinant is simply the product of its diagonal elements. This is because the determinant is a measure of volume (or scale factor in transformations) and, in these matrices, all the 'non-effective' zeros simplify this calculation.
• Determinant of the Identity Matrix: If all elements on the diagonal are 1 (as is the case in an identity matrix, which is also a triangular matrix), then the determinant is also 1. This holds true for both upper and lower triangular matrices.
• Matrix Inverse and Nullity: The determinant being zero means the matrix is singular, hence it does not have an inverse. In triangular matrices, this happens only if at least one diagonal element is zero.

Leveraging these properties simplifies many matrix operations and helps solve systems of linear equations more efficiently.