Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

If $$ A=\left[\begin{array}{rrrr} a_{11} & a_{12} & a_{13} & a_{14} \\ 0 & a_{22} & a_{23} & a_{24} \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & 0 & a_{44} \end{array}\right] $$ show that \(|A|=a_{11} a_{22} a_{33} a_{44}\)

Step-by-Step Solution

Answer

The determinant \(|A|\) is the product of its diagonal elements: \(a_{11} a_{22} a_{33} a_{44}\).

Step 1: Understand the Matrix Type

The given matrix \(A\) is an upper triangular matrix, which means all elements below the main diagonal are zero.

Step 2: Recall the Determinant Property of Upper Triangular Matrices

For an upper triangular matrix, the determinant is the product of the elements on its main diagonal.

Step 3: Identify the Main Diagonal Elements

In the matrix \(A\), the main diagonal elements are \(a_{11}, a_{22}, a_{33},\) and \(a_{44}\).

Step 4: Calculate the Determinant

By applying the property from Step 2, the determinant of matrix \(A\) is \(|A| = a_{11} \cdot a_{22} \cdot a_{33} \cdot a_{44}\).

Step 5: Conclusion

Thus, we have shown that \(|A| = a_{11} a_{22} a_{33} a_{44}\), confirming the solution.

Key Concepts

Triangular Matrices

Triangular matrices are a special type of square matrix, known for their distinctive structure. They come in two varieties: upper triangular and lower triangular matrices. In an upper triangular matrix, all elements below the main diagonal are zero. This creates a triangular shape, hence the name.
For example, consider an upper triangular matrix:

• The elements above and on the main diagonal can be any value.
• All elements below the main diagonal are zero.

In contrast, a lower triangular matrix has all elements above the main diagonal set to zero. Understanding these types of matrices is crucial, especially when working with determinants, as they simplify calculations significantly.

Properties of Determinants

Determinants possess some fascinating properties that are helpful in matrix-related calculations. Knowing these properties can save time, especially with special matrices like triangular matrices.

• Multiplicative Property: For two matrices, the determinant of their product equals the product of their determinants: \[ |AB| = |A| \cdot |B| \]
• Row and Column Swap: Swapping two rows or columns of a matrix changes the sign of its determinant.
• Zero Row or Column: If a matrix has a row or column entirely of zeros, its determinant is zero.
• Triangular Matrices: The determinant of a triangular matrix—either upper or lower—is simply the product of the elements on its main diagonal.

These properties, especially the one concerning triangular matrices, are pivotal in determining \( |A| = a_{11} \cdot a_{22} \cdot a_{33} \cdot a_{44} \) for an upper triangular matrix.

Main Diagonal Elements

In a square matrix, the main diagonal is an array of elements directly proceeding from the upper left corner to the lower right corner of the matrix.

• It includes elements like \(a_{11}, a_{22}, a_{33}, \ldots\).
• These elements are critical as they often simplify complex matrix operations.

For triangular matrices, these main diagonal elements hold special significance. The determinant of such matrices is simply the product of these diagonal elements. So, when given a triangular matrix, locate these main diagonal values first, as they provide a direct calculation path to find the determinant.
In our example matrix, with elements \(a_{11}, a_{22}, a_{33}, a_{44}\) forming the main diagonal, the determinant can be calculated with ease. This characteristic greatly streamlines the computation process for determinants, particularly with triangular matrices.