Problem 69 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is -168.

Step 1: Identify the type of matrix

Identify that the given matrix is a special type called a triangular matrix. In a triangular matrix, all the entries above the main diagonal (for a lower triangular matrix) or below the main diagonal (for an upper triangular matrix) are zero.

Step 2: Apply the rule for the determinant of a triangular matrix

Apply the rule that the determinant of a triangular matrix is the product of the entries on its main diagonal. In other words, multiply all the diagonal entries together.

Step 3: Solve for the determinant

The diagonal entries of the given matrix are -6, -1, -7, -2, -2. Multiply these together to compute the determinant: \((-6) \times (-1) \times (-7) \times (-2) \times (-2) = -168\).

Key Concepts

Triangular Matrix

A triangular matrix is a specific type of square matrix where either all the elements above the main diagonal are zero, making it a lower triangular matrix, or all elements below the main diagonal are zero, creating an upper triangular matrix. This structure simplifies many matrix operations, such as finding the determinant.

Key features to remember about triangular matrices:

• Lower triangular matrices have non-zero elements on and below the diagonal.
• Upper triangular matrices have non-zero elements on and above the diagonal.
• They are prevalent in numerical methods and solving linear equations.
• The simplicity in their structure helps reduce complexity in calculations.

Recognizing a triangular matrix is the first step in many mathematical operations. Once identified, certain computational rules, like calculating the determinant or performing Gaussian elimination, become simpler tasks.

Diagonal Matrix

A diagonal matrix is a more specific form of a triangular matrix. In a diagonal matrix, all the entries outside the main diagonal are zero. This means it is both an upper and lower triangular matrix simultaneously.

Useful properties of diagonal matrices include:

• Determinant calculation is straightforward: it is the product of the diagonal entries.
• The inverse of a diagonal matrix (if it exists) is also diagonal and simply involves taking the reciprocal of each non-zero diagonal element.
• Matrix multiplication when involving a diagonal matrix is highly efficient due to the zeros outside the main diagonal.

Due to their simple structure, diagonal matrices often serve as effective preconditioners in numerical methods and make certain algebraic operations much easier to perform.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, involving the combination of rows and columns between matrices. While straightforward when dealing with matrices of the same dimensions, it can seem complex when the structures vary.

Guidelines for matrix multiplication:

• To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second.
• The resulting matrix takes the number of rows from the first and columns from the second matrix.
• Each entry in the product matrix is calculated by multiplying corresponding entries from the row of the first matrix and the column of the second and summing them.

When a triangular or diagonal matrix is involved in multiplication, computations can be more manageable due to their zero-filled structure. For instance, multiplying by a diagonal matrix involves scaling rows or columns without complex summing, streamlining many matrix operations significantly.