Problem 43 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} -1 & 8 & -3 \\ 0 & 3 & -6 \\ 0 & 0 & 3 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is -9

Step 1: Identifying the Matrix Type

First, we differentiate between the square matrices. This matrix is identified as a lower triangular matrix since all entries above the main diagonal are zero.

Step 2: Calculating the Determinant for Lower Triangular Matrix

For a lower triangular matrix, determinant is the product of the elements on the main diagonal. Therefore, we calculate the determinant as the product of -1, 3, and 3.

Step 3: Solving the Calculation

The determinant is -1*3*3= -9

Key Concepts

Triangular Matrix

In linear algebra, a triangular matrix is a special type of square matrix that simplifies many matrix operations, especially determinant calculations. There are two types of triangular matrices: upper triangular and lower triangular.
Upper triangular matrices have all elements below the main diagonal as zero. In contrast, lower triangular matrices have all elements above the main diagonal as zero.
A key property of triangular matrices is that the determinant is simply the product of the diagonal elements. This is especially useful for simplifying computations, as seen in our example exercise.

• For upper triangular matrices, the determinant can be found by multiplying all the elements that lie on the diagonal starting from the top left corner.
• For lower triangular matrices, similar logic applies, but you start from the bottom left corner.

Understanding these properties makes it easier to solve complex matrix problems quickly and efficiently.

Cofactor Expansion

Cofactor expansion is a method used to find the determinant of a matrix, particularly useful when matrices are neither triangular nor easily reducible. This process involves using minors and cofactors.
To perform a cofactor expansion, we select a row or column (often the one with the most zeros to minimize complexity). The determinant is calculated by expanding across this row or column.
The key steps are:

• Identify a row or column to expand across.
• Calculate the minor for each element, which is the determinant of the sub-matrix formed when removing the row and column of that element.
• Multiply each minor by the element from the matrix and the correct sign (+/-), based on the position's cofactor (alternating signs).
• Sum these values to find the determinant.

This method is versatile and can be more computationally intensive than using properties of triangular matrices, but it is fundamental for understanding determinants.

Matrix Operations

Matrix operations are fundamental tools in linear algebra, essential for solving systems of equations, transformations, and more. Key operations include addition, subtraction, multiplication, and finding determinants.
For a given matrix, the determinant provides important information about the matrix, such as whether a matrix is invertible. Invertibility is crucial for solving systems of linear equations using matrices.
Here are essential matrix operations you need to know:

• Addition and Subtraction: Possible when matrices have the same dimensions. Add or subtract corresponding elements.
• Multiplication: Involves the dot product of rows and columns. This can be computationally demanding but is necessary for many applications.
• Finding the Determinant: Provides a scalar value that summarizes certain matrix properties, like singularity or invertibility.

Being comfortable with matrix operations, including determinants and triangular matrices, equips you to tackle more complex problems in mathematics and applied sciences.