Problem 54 - Algebra for College Students

Question

Consider the following matrix: $$ A=\left[\begin{array}{rrrrrr} 4 & 3 & 2 & 1 & 5 & -3 \\ 5 & 2 & 7 & 8 & 6 & 3 \\ 0 & 9 & 1 & 4 & 7 & 2 \\ 4 & 3 & 2 & 1 & 5 & -3 \\ -4 & -6 & 7 & 12 & 11 & 9 \\ 5 & 8 & 6 & -3 & 2 & -1 \end{array}\right] $$ Use your calculator to show that \(|A|=0\).

Step-by-Step Solution

Answer

The determinant of matrix \( A \) is 0, indicating it's singular.

Step 1: Finding Determinant Configuration

The determinant of a matrix can only be calculated if the matrix is square, meaning it has an equal number of rows and columns. Matrix \( A \) is a \( 6 \times 6 \) matrix, which is square and therefore eligible for determinant calculation.

Step 2: Utilizing Calculator for Determinant Computation

On your scientific calculator, access the matrix feature. Input the matrix \( A \) row by row. Check that the matrix is entered correctly by comparing it with the original matrix.

Step 3: Performing Determinant Calculation

With the matrix entered, locate the determinant calculation option, often labeled as \('det'\), and apply it to the matrix \( A \). This operation will compute the determinant value.

Step 4: Interpret Result

After you compute the determinant through your calculator, observe the result. In this case, it should display that the determinant \(|A| = 0\). The result confirms a singular matrix, which has no inverse.

Key Concepts

Square Matrix

A square matrix is a type of matrix that has the same number of rows and columns. In simpler terms, if you look at a matrix and notice that it’s perfectly symmetrical, like a square shape on a grid, it's a square matrix. For instance, a 6x6 matrix has 6 rows and 6 columns. It’s crucial to know if a matrix is square because only square matrices allow us to calculate what is known as the "determinant".
A matrix being square is one of the essential criteria for exploring various mathematical properties such as finding the inverse of a matrix or performing certain transformations. So when you’re dealing with matrices, always check the dimensions first to see if it's square. If they're equal, you're good to go for the next steps, including determinant calculation.

Determinant Calculation

Calculating the determinant of a square matrix is a fundamental task in linear algebra. The determinant is a special number that gives us important information about the matrix.

• First, the determinant helps us determine whether a matrix is invertible, meaning it has an inverse.
• Second, it provides insights about the linear independence of the matrix's columns.

To calculate the determinant of a 6x6 matrix like matrix \( A \), it's typically more practical to use a calculator due to its complexity. Here’s how you can do it:

• Enter the matrix into your calculator, ensuring each row and column matches the original.
• Find the function, usually labeled as 'det', which computes the determinant.

This process is straightforward with technology, but understanding why we calculate the determinant is what truly enriches your mathematical knowledge. A non-zero determinant implies possible further explorations, while a zero determinant, like in this example, tells us the matrix is singular.

Singular Matrix

A matrix is described as 'singular' when its determinant is zero. This is an important mathematical property with several implications.

• Firstly, a singular matrix like matrix \( A \) does not have an inverse. This means you cannot "undo" the transformations it represents, which can be a critical factor in solving systems of equations.
• A determinant of zero also means that the transformation matrix \( A \) represents collapses the space into a lower dimension. The rows or columns of the matrix are in some way redundant or linearly dependent.

Understanding singular matrices is key in multiple fields where linear algebra is applied, such as computer graphics, engineering, and beyond. When tackling problems, if you calculate the determinant and find it to be zero, you immediately know that the matrix cannot be used for operations requiring an inverse.