Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{ll}4 & 2 \\ 6 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is 0, so the inverse does not exist.

Step 1: Calculate the determinant

Determinant formula for a 2x2 matrix \(\left[\begin{array}{ll}a & b\\\ c & d\end{array}\right]\) is: \[D = ad - bc\] For the given matrix \(\left[\begin{array}{ll}4 & 2\\\ 6 & 3\end{array}\right]\): \[D = 4\times3 - 6\times2\]

Step 2: Check if the determinant is zero

\[D = 12 - 12 = 0\] Since the determinant is 0, the inverse of the given matrix does not exist. Therefore, there is no need to proceed with the next steps or to verify the answer.

Key Concepts

Determinant Calculation

Determining the determinant of a matrix is essential in finding out if the matrix has an inverse. For a 2x2 matrix like our example, the formula used to calculate the determinant is:

• Given a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\], the determinant \ \ \(D\) can be computed as \(ad - bc\).

In our example, the matrix\(\begin{bmatrix} 4 & 2 \ 6 & 3 \end{bmatrix}\),the determinant \ \(D\) is calculated by multiplying \((4 \times 3)\) and subtracting \((6 \times 2)\):\[D = 4 \times 3 - 6 \times 2 = 12 - 12 = 0\]This simple calculation helps identify whether the matrix is invertible.

2x2 Matrix

A 2x2 matrix is a foundational element in matrix algebra. It is a matrix with two rows and two columns. Given by:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\], it's characterized by the direct positioning and order of its elements.

• Matrix elements \(a, b, c,\) and \(d\) are crucial when performing operations such as addition, multiplication, and finding determinants.

Given its simplicity, 2x2 matrices are often used as introductory subjects for understanding more complex matrix operations. Operations involving these matrices form the basis for more advanced concepts in linear algebra, including eigenvalues, eigenvectors, and transformations.

Inverse Existence

The existence of an inverse for a matrix is contingent upon its determinant. Specifically for a 2x2 matrix, an inverse exists only if the determinant is not zero. If the determinant \ \(D\)\ is calculated to be zero, it means:

• There is no matrix that when multiplied by the original matrix would result in an identity matrix.
• The matrix is termed as singular and thus non-invertible.

In our example, as calculated earlier, the determinant was found to be zero. Hence, there is no inverse for this matrix. This property is a vital check in linear equations for solving whether unique solutions can be determined from the systems they represent.

Matrix Algebra

Matrix algebra encompasses a range of operations that can be performed on matrices to solve systems of equations and transform geometric entities. Fundamental operations include addition, subtraction, and multiplication. Determining the inverse of a matrix is an operation typically seen where:

• An inverse is used to solve linear equations of the form \(AX = B\), where \(A^{-1}\) is sought to isolate \(X\).

Matrix multiplication and the computation of inverses (when they exist) showcase the usefulness of matrices in applications ranging from computer graphics to solving dynamic systems. Mastery of these operations forms the backbone of many practical fields in science and engineering, underlining the importance of matrix algebra in both theory and application.