Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 6 & -5 & 0 & 0 \\ 1 & 3 & 2 & 0 \\ 1 & 2 & 7 & -1 \end{array}\right] $$

Short answer

The determinant of the matrix is \( 4 \times -5 \times 2 \times -1 = 40 \)

Step-by-step solution

Identify the diagonal elements

The diagonal elements of the given matrix are 4, -5, 2, and -1. These are the terms from the upper left to the lower right of the matrix.

Compute the determinant

The determinant of a triangular matrix is the product of its diagonal elements. Therefore, calculate: \( 4 \times -5 \times 2 \times -1 \)

Key concepts

Triangular Matrices

A triangular matrix is a special form of a square matrix where all the elements above or below the diagonal are zero. There are two types of triangular matrices: upper triangular and lower triangular. In an upper triangular matrix, all the elements below the diagonal are zero, while in a lower triangular matrix, all the elements above the diagonal are zero.

• **Upper Triangular Matrices:** These matrices have zeros below their main diagonal. An example is the matrix given in the exercise, where you can see that all elements below the diagonal are non-zero, suggesting it is not a typical upper triangular matrix.
• **Lower Triangular Matrices:** Conversely, these matrices have zeros above their main diagonal.

Understanding the concept of triangular matrices is crucial because it greatly simplifies matrix operations, including finding determinants. The matrix from the exercise is a lower triangular matrix, which makes it much easier to compute its determinant than a non-triangular matrix.

Diagonal Elements

The diagonal elements of a matrix are those positioned from the top left to the bottom right. In a square matrix, these elements are particularly important in various calculations, such as when determining the determinant of triangular matrices. For instance, the given matrix in the exercise has diagonal elements 4, -5, 2, and -1. These are the numbers located at positions (1,1), (2,2), (3,3), and (4,4) respectively.

• **Why are Diagonal Elements Important?** In triangular matrices, the determinant is the simple product of these diagonal elements, which is not the case in general square matrices. This property makes calculations more straightforward.
• **Special Properties:** In identity matrices, the diagonal elements are all 1, which helps them function as the multiplicative identity in matrix multiplication.

Matrix Multiplication

Matrix multiplication is a way of multiplying two matrices by taking the dot product of rows and columns. However, when calculating the determinant of a triangular matrix, the process involves multiplying only the diagonal elements. This is a special exception to the usual rules of matrix multiplication.

• **Standard Matrix Multiplication:** Involves taking each element of a row from the first matrix and each element of a column from the second matrix and summing their products. This process is carried out for every row and column pair.
• **Simplified for Determinants:** In the case of triangular matrices, we don't need to go through the standard tedious multiplication. For example, in the exercise, to find the determinant, you simply multiply the diagonal elements: 4, -5, 2, and -1. Therefore, the determinant is calculated as \(4 \times -5 \times 2 \times -1 = 40\).

Understanding matrix multiplication and its simplifications in specific cases like triangular matrices helps in efficiently solving matrix-related problems.