Problem 13 - College Algebra with Modeling and Visualization

Question

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of matrix \(A\) is 30.

Step 1: Identify the Matrix and Recall Definition

We start with matrix \(A\) given as \(\begin{bmatrix} 2 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 5 \end{bmatrix}\). The determinant of a 3x3 diagonal matrix \( \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} \) is directly equal to the product of its diagonal elements \( a \cdot b \cdot c \).

Step 2: Apply the Definition of Determinant for Diagonal Matrices

Since the matrix \(A\) is a diagonal matrix, we can find the determinant by multiplying its diagonal elements. Thus, \( \text{det}(A) = 2 \times 3 \times 5 \).

Step 3: Calculate Det(A)

Perform the multiplication of the diagonal elements: \( 2 \times 3 = 6 \) and then \( 6 \times 5 = 30 \). So, the determinant \( \text{det}(A) = 30 \).

Step 4: Conclusion

The determinant of the matrix \(A\) is 30. There was no need to expand by cofactors since it's a diagonal matrix.

Key Concepts

Diagonal Matrices

A diagonal matrix is a special kind of square matrix. It simplifies many calculations in linear algebra because all non-diagonal elements are zero. This means it's presented in a format where everything outside its main diagonal is zero, like the matrix from the original exercise:

• The elements along the diagonal are called diagonal elements, and in a 3x3 matrix, these are placed at positions (1,1), (2,2), and (3,3).
• Diagonal matrices are simple to work with, as their determinant is just the product of these diagonal elements.

For example, if you have \[\begin{bmatrix} a & 0 & 0 \0 & b & 0 \0 & 0 & c \end{bmatrix},\]the determinant would be \(a \times b \times c\).
This property makes calculating determinants of diagonal matrices very efficient. Instead of going through complex cofactor expansions, you simply multiply the diagonal elements.

Cofactor Expansion

The concept of cofactor expansion is key for determining the determinant of a matrix, especially when you're not dealing with diagonal matrices. It involves breaking down larger matrices into smaller ones, focusing on minors and signatures. Here's a rundown of what happens:

• Choose a row or column. This affects the computation method but not the end result.
• Calculate the minor for each element in this row or column by removing the respective row and column, simplifying your work to a smaller matrix.
• Multiply each minor by the element in the original matrix and the corresponding sign from the sign pattern (+, -, +).

This expansion works well with any square matrix but becomes computationally intense as the matrix size grows. Luckily, if you're working with a diagonal or other special matrices, simpler rules might be more applicable, as in our exercise example.

Determinant Calculation

Determinants are an important aspect of linear algebra, offering insight into matrix properties such as invertibility or eigenvalues. Usually, the determinant is calculated using cofactor expansion or, in special cases like diagonal matrices, by straightforward multiplication of elements from the main diagonal. When calculating determinants:

• Check if the matrix is special, like diagonal or triangular. This often allows for simple computation.
• For regular matrices, apply cofactor expansion if necessary.
• Carry out multiplications and any necessary calculations sequentially to prevent errors.

Determinants can sometimes be zero, indicating that a matrix is singular (non-invertible). Understanding how to efficiently calculate determinants equips you to tackle many linear algebra situations, determining crucial properties of matrices quickly and effectively.