Problem 109 - Precalculus

Question

Why does the square matrix \(A=\left[\begin{array}{ll}2 & 3 \\ 4 & 6\end{array}\right]\) not have an inverse?

Step-by-Step Solution

Answer

Matrix \( A \) does not have an inverse because its determinant is 0.

Step 1: Define the Determinant Criterion for Invertibility

A square matrix is invertible (also called non-singular) if its determinant is non-zero. Therefore, to determine if the matrix \( A \) has an inverse, we need to calculate its determinant.

Step 2: Compute the Determinant of the Matrix

The determinant of a 2x2 matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) is calculated using the formula \( ad - bc \). For the given matrix \( A = \left[\begin{array}{cc} 2 & 3 \ 4 & 6 \end{array}\right] \), we have:\[\text{det}(A) = (2)(6) - (3)(4) = 12 - 12 = 0\]

Step 3: Conclude on the Invertibility Based on the Determinant

Since the determinant of matrix \( A \) is 0, the matrix is singular and thus does not have an inverse. A determinant of 0 means that the rows or columns of the matrix are linearly dependent, further indicating non-invertibility.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that gives us important information about the matrix. In the context of a 2x2 matrix, it's a simple calculation that tells us, among other things, whether the matrix is invertible or not. For any 2x2 square matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed as \( ad - bc \).
In simpler terms, it's like a special number that we derive from the matrix by multiplying and then subtracting products of its diagonals.

• If the determinant is not zero, the matrix can be inverted, meaning there's another matrix that can "undo" its effect when multiplied.
• If the determinant is zero, the matrix cannot be inverted, and this is a crucial finding.

The determinant provides a quick check to establish these key properties of the matrix.

Singular Matrix

A singular matrix is one term that comes up when discussing matrix invertibility—or rather, lack thereof. A matrix is singular, meaning it does not have an inverse, when its determinant is zero. This is a big red flag that tells us something is amiss with how the matrix behaves. For example, if we calculate the determinant of a matrix A, and it turns out to be zero, that's when we've identified that the matrix is singular.
Being singular doesn't just hint at non-invertibility, but it also implies some structural dependencies within the matrix itself. This deficiency in having an inverse means we can't solve certain equations using this matrix, something very important when solving systems of linear equations.
Hence, calling a matrix singular paints a broad picture about its utility and functionality in computations.

Linear Dependence

Linear dependence is a concept tied to the behavior of matrices and their potential invertibility. When discussing a matrix whose determinant is zero, we often describe its rows or columns as being linearly dependent.
Simply put, linear dependence occurs when at least one row or column in the matrix can be expressed as a combination of others. For example, if you can multiply one row by a number and add to another row to get zero, they are dependent. This relationship is why the matrix's determinant is zero.

• It means there's some redundancy in the matrix, as not all rows or columns are providing unique information.
• This zero-determinant phenomenon makes it impossible for the matrix to be invertible.

Recognizing linear dependence tells us much about why a matrix can't be inverted, thereby helping us understand its limits in solving problems.