Problem 16 - Advanced Engineering Mathematics

Question

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is -30.

Step 1: Understand the Matrix

The given matrix is a 3x3 diagonal matrix: \( \begin{pmatrix} 5 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 2 \end{pmatrix} \). Diagonal matrices have non-zero elements only on their main diagonal (from top left to bottom right).

Step 2: Recall the Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is simply the product of its diagonal elements. Therefore, for a matrix \( \begin{pmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{pmatrix} \), the determinant is \( a \cdot b \cdot c \).

Step 3: Calculate the Determinant

Multiply the diagonal elements together: \[ 5 \cdot (-3) \cdot 2 = -30 \].The determinant of the matrix is -30.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements outside the main diagonal are zeros. In simpler terms, only the numbers from the top left corner to the bottom right corner are non-zero. It's easy to spot a diagonal matrix because of its shape and the placement of zeros. Some characteristics of diagonal matrices include:

• The elements on the main diagonal can be any number, including zero, but all off-diagonal elements must be zero.
• Diagonal matrices are symmetric if the non-zero elements on the diagonal are the same when flipped across the main diagonal.
• These matrices are simple to work with as operations like addition and multiplication are straightforward due to the zeros in non-diagonal positions.

For students, understanding diagonal matrices and their properties can significantly simplify solving linear algebra problems.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a matrix. This technique is particularly useful for matrices that are not in diagonal form, as it allows you to break down a larger determinant into smaller, more manageable pieces. Here's how cofactor expansion works:

• Choose a row or a column of the matrix. It's often easiest to pick a row or column with zeros, which minimizes computation.
• For each element in the chosen row or column, remove the row and column that contain that element. The remaining matrix is the minor of that element.
• Calculate the determinant of each minor matrix.
• Multiply each determinant by the original element and its corresponding sign, as given by the checkerboard pattern of signs. The sign is positive for even-positioned elements (e.g., top-left) and negative for odd-positioned elements.
• Add these values together to get the determinant of the original matrix.

While cofactor expansion is a powerful tool, remember that for diagonal matrices, this method simplifies vastly, as only the main diagonal elements need to be considered.

3x3 Matrix Determinant

Calculating the determinant of a 3x3 matrix is an essential skill in linear algebra. The determinant provides information about the matrix, such as whether it is invertible or singular. In a 3x3 matrix like our example, understanding its determinant can also give insight into the vector space properties related to the matrix.For a general 3x3 matrix \[\begin{bmatrix}a & b & c \d & e & f \g & h & i \end{bmatrix},\]the determinant is calculated using:\[a(ei - fh) - b(di - fg) + c(dh - eg).\]However, if the matrix is a diagonal matrix, the determination process becomes even easier. Instead of calculating using the formula above, simply multiply the diagonal elements together, which for a matrix \[\begin{bmatrix}a & 0 & 0 \0 & b & 0 \0 & 0 & c \end{bmatrix}\]results in a determinant of \( a \times b \times c \). This simplification is what makes diagonal matrices so efficient in simplifying determinant calculations. Understanding this concept helps avoid unnecessary steps and encourages more efficient problem-solving approaches.