Problem 13 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix A is not invertible because its determinant is 0.

Step 1: Determinant Calculation

The formula to find the inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). First, we need to find the determinant of matrix \(A\).The determinant \(det(A)\) is calculated as follows: \(-6 \cdot 2 - (-3) \cdot 4 = -12 + 12 = 0\).

Step 2: Check for Invertibility

A matrix is only invertible if its determinant is not equal to zero. Since the determinant of matrix \(A\) is 0, \(A\) does not have an inverse.

Key Concepts

The Determinant of a Matrix

In the context of matrices, the determinant is a special scalar value that can provide a lot of information about the properties of a matrix. For a 2x2 matrix given by \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\] the determinant, denoted as \( \det(A) \), is calculated using the formula:\[\det(A) = ad - bc.\]

• The determinant helps us understand whether a matrix is invertible.
• If the determinant equals zero, the matrix is singular and does not have an inverse.
• If the determinant is non-zero, the matrix is non-singular, implying it has an inverse.

For example, consider the matrix \[A = \begin{bmatrix} -6 & 4 \ -3 & 2 \end{bmatrix}.\]Calculating its determinant gives \[(-6)(2) - (-3)(4) = -12 + 12 = 0.\]Thus, the determinant is zero, indicating that this particular matrix is not invertible.

Understanding the 2x2 Matrix

A 2x2 matrix is a grid that contains two rows and two columns filled with numbers. It is one of the simplest forms of a matrix, making it a great starting point for learning matrix operations like finding an inverse. The general form of a 2x2 matrix is written as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}.\]

• Each component of the matrix, \(a, b, c,\) and \(d\), is a real number that affects the matrix’s properties.
• Arithmetic operations, like addition, multiplication, and finding the determinant and inverse, become intuitive once you know these elements.
• These matrices are extensively used in transformations, systems of equations, and linear algebra.

The simplicity of the 2x2 matrix showcases fundamental linear algebra concepts, providing a manageable way to compute determinants and inverses.

Invertibility and Its Significance

The concept of invertibility in matrices ties closely with solving systems of equations and ensuring transformations are reversible. A matrix is invertible if and only if it has an inverse, meaning that there exists another matrix which, when multiplied together, results in an identity matrix.

• An identity matrix in the 2x2 form is \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix},\]serving as a multiplicative equivalent of the number 1 in basic arithmetic.
• Invertibility hinges entirely on the determinant: only non-zero determinants allow for an inverse.
• If a matrix is not invertible, it is considered singular. Operations like matrix division become impossible, making the matrix limited in its applications.

Understanding whether a matrix is invertible is thus crucial in fields such as engineering, physics, and computer graphics, where matrix operations model real-world data.