Problem 14 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} 5 & 10 \\ -3 & -6 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \(A\) does not exist because its determinant is zero.

Step 1: Check if the inverse exists

To find the inverse of a matrix, we first need to check if the determinant of matrix \(A\) is non-zero. The formula for the determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\). For matrix \(A\):\[\det(A) = (5)(-6) - (10)(-3) = -30 + 30 = 0.\]Since the determinant is 0, matrix \(A\) does not have an inverse.

Step 2: Conclusion

Since the determinant of matrix \(A\) is zero, matrix \(A\) is singular, meaning \(A^{-1}\) does not exist.

Key Concepts

Determinant

The determinant is a special number that you can calculate directly from a square matrix. It is a scalar that gives important information about the matrix, such as whether the matrix is invertible or not.
To determine this number for a 2x2 matrix, use the formula:

• If the matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then the determinant, denoted as \( \det(A) \), is calculated as \( ad - bc \).

In the context of solving matrices, one of the most crucial uses of the determinant is to check if a matrix is invertible. A non-zero determinant indicates that an inverse does exist.
For instance, if we have matrix \( A = \begin{bmatrix} 5 & 10 \ -3 & -6 \end{bmatrix} \), the determinant would be:

• \( \det(A) = (5)(-6) - (10)(-3) = -30 + 30 = 0 \)

In this case, the determinant equals zero, implying that the matrix is not invertible. This is a crucial concept when solving matrix-related problems because if the matrix cannot be inverted, certain calculations or solutions, such as systems of linear equations, cannot proceed with this matrix as planned.

Singular Matrix

A singular matrix is a matrix that does not have an inverse. This typically occurs when the determinant of the matrix is zero. This concept is crucial when dealing with matrices, especially in linear algebra and computational mathematics.
Understanding why a matrix is singular involves calculating its determinant. If the determinant equals zero, it means there is no unique solution when solving equations involving this matrix. Such matrices represent equations or transformations that do not have full rank or coverage, meaning they collapse into lower dimensional spaces or do not span the full space they represent.
In the exercise, the given matrix \( A = \begin{bmatrix} 5 & 10 \ -3 & -6 \end{bmatrix} \) has a calculated determinant of zero.

• This zero determinant leads to it being classified as singular.

Therefore, since it is a singular matrix, it does not have an inverse. Recognizing singular matrices is vital for problem-solving and ensuring that no further incorrect assumptions are made during calculations.

2x2 Matrix

A 2x2 matrix is one of the simplest types of matrices used in mathematics and physics. It consists of two rows and two columns. Despite its simplicity, a 2x2 matrix can exhibit a wide range of properties and is often used to illustrate basic concepts of matrices, such as inversion, and determinant computation.
Here’s a general depiction of a 2x2 matrix:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

Each element \( a, b, c, \) and \( d \) is a component of the matrix. To find an inverse of a 2x2 matrix, it must first be determined that the determinant is non-zero.

• The formula for the inverse, if it exists, is \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

However, if \( ad-bc = 0 \), as it is with this exercise, the formula breaks down because you would be dividing by zero. This impossibility is a direct consequence of the determinant being zero, showing once again the matrix is non-invertible.
Understanding 2x2 matrices is helpful because they form the foundation for larger matrices and more complex operations in linear algebra.