Problem 46 - Algebra 2

Question

OPEN ENDED Create a square matrix that does not have an inverse. Explain how you know it has no inverse.

Step-by-Step Solution

Answer

Matrix \( A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) is singular because its determinant is zero, so it has no inverse.

Step 1: Understanding Inverse of a Matrix

A square matrix has an inverse if and only if it is nonsingular, which means its determinant is non-zero. If a matrix is singular (determinant equals zero), it does not have an inverse.

Step 2: Choosing a Simple Matrix

For simplicity, let's choose a 2x2 matrix because calculations are straightforward, and concepts can still be demonstrated clearly. We'll consider the matrix: \[ A = \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix} \]

Step 3: Calculating the Determinant

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). For our matrix, the calculation is:\[\det(A) = 1 \cdot 4 - 2 \cdot 2 = 4 - 4 = 0.\] Since the determinant is zero, this matrix does not have an inverse.

Step 4: Conclusion

Matrix \( A = \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix} \) is singular because its determinant is zero. Therefore, it does not have an inverse.

Key Concepts

Singular Matrix

A **singular matrix** is an important concept in linear algebra. It refers to a square matrix that does not have an inverse. This situation occurs when the matrix's determinant is zero.
For example, consider the matrix \( A = \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix} \). The determinant of this matrix is zero, as calculated by \( ad - bc = 1 \times 4 - 2 \times 2 = 0 \).
Because the determinant is zero, matrix \( A \) is singular. This means there is no other matrix you can multiply it by to get the identity matrix, a property that defines invertible matrices. In simpler terms, a singular matrix "collapses" in such a way that it loses its ability to "stretch" back to its original shape, i.e., it can't be reversed.

Determinant

The **determinant** is a special number that can be calculated from a square matrix. It provides critical insights, such as whether or not the matrix has an inverse.
For a 2x2 square matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed by the formula \( ad - bc \).
The value of the determinant tells us a lot:

• If it's zero, the matrix is singular and does not have an inverse.
• If it's non-zero, the matrix is nonsingular and has an inverse.

In essence, the determinant acts as a flag, alerting us to the matrix's properties concerning invertibility.

Square Matrix

A **square matrix** is a matrix with the same number of rows and columns. This type is particularly significant because only square matrices can have inverses.
Matrices like \( 2 \times 2 \), \( 3 \times 3 \), etc., are all square matrices.
This characteristic lays the foundation for more advanced operations, such as determining whether a matrix is invertible or finding solutions to linear equations.
Thus, a square shape is not just a simple geometric pattern; it's a prerequisite for exploring deeper mathematical properties like eigenvalues and determinants.

Nonsingular Matrix

A **nonsingular matrix** is a square matrix that does have an inverse. For this to be the case, the matrix's determinant must be non-zero.
To understand this, consider the computation: if a matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and its determinant is \( ad - bc eq 0 \), then it is nonsingular.

• The absence of a zero determinant ensures that there is a unique inverse matrix.
• It signifies that the matrix can "stretch" and "return" to its identity shape, confirming its invertibility.

A nonsingular matrix is robust and full of flexibility, paving the path for manipulating and solving sets of linear equations efficiently.