Problem 29 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} 2 & 1 & 2 \\ 5 & 10 & 5 \\ 3 & 6 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \(A\) does not have an inverse because it is singular.

Step 1: Verify the matrix is square

A matrix must be square to have an inverse. Matrix \(A\) is a \(3 \times 3\) matrix which is square, so it is eligible for an inverse calculation.

Step 2: Calculate the determinant of A

The determinant of matrix \(A\) is necessary to determine if an inverse exists. Calculate the determinant using the formula for a 3x3 matrix: \[\det(A) = 2(10 \cdot 3 - 6 \cdot 5) - 1(5 \cdot 3 - 5 \cdot 3) + 2(5 \cdot 6 - 10 \cdot 3) \]Simplifying, we have:\[\det(A) = 2(30 - 30) - 1(15 - 15) + 2(30 - 30)\]\[\det(A) = 2(0) - 1(0) + 2(0) = 0\] The determinant is 0.

Step 3: Determine if the inverse exists

Since the determinant of \(A\) is 0, this means the matrix \(A\) is singular. A singular matrix does not have an inverse.

Key Concepts

Determinant Calculation

The determinant of a matrix plays a crucial role when determining if a matrix has an inverse. For a 3x3 matrix, the formula can seem a bit daunting, but it's not too bad once you break it down. The determinant helps us know if a matrix can be reversed, kind of like figuring out if a path has a way back.

To calculate the determinant of a 3x3 matrix, you use the following method:

• Multiply across the diagonals of the matrix. You'll do this a few times over, following what's known as the cofactor expansion method.
• Subtract and add these products accordingly to get the final determinant value.

This process is straightforward with practice and is critical in matrix operations. If a determinant is zero, the matrix is crucially singular, meaning it doesn't have an inverse.

Singular Matrix

A singular matrix is a term used for matrices that do not have an inverse. Imagine you're working with numbers: dividing by zero is a big no-no. Similarly, when a matrix is singular, it's basically the matrix version of a non-invertible situation.

There are a few tell-tale signs of a singular matrix:

• The determinant is zero, as you've seen in the problem here.
• The rows or columns might be multiples of each other, making them linearly dependent.
• It doesn't span the whole space, which you sometimes can visualize in geometric terms.

When dealing with such matrices, it's important to notice these characteristics early on. Knowing a matrix is singular helps in understanding why it cannot simply "go backwards," unlike regular, invertible matrices.

3x3 Matrix

The 3x3 matrix is a foundational building block in linear algebra. These matrices help in representing systems of linear equations with up to three variables. A 3x3 matrix is simple enough to be understandable and manually computable but complex enough to demonstrate key matrix properties.

Key points to remember about 3x3 matrices:

• They are used in various applications, such as rotations in 3D graphics and modeling systems of equations.
• Each entry in the matrix is essential for calculations including determinants and potential inversion.
• A 3x3 matrix makes the perfect basis for understanding larger matrices.

In your exercise, you work with these to explore inversions—but it's vital to understand the groundwork first. With each matrix, there's a pattern or rule that governs its operations and usability.