Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix is \( \left[ \begin{array}{cc} 0.8 & -0.6 \ -0.2 & 0.4 \end{array} \right] \).

Step 1: Calculate the determinant

The determinant of the given 2x2 matrix \( \left[ \begin{array}{cc} 2 & 3 \ 1 & 4 \end{array} \right] \) is calculated as \( (2*4) - (3*1) = 5 \).

Step 2: Swap 'a' and 'd'

Swap the positions of 'a' (2) and 'd' (4) in the matrix and it becomes \( \left[ \begin{array}{cc} 4 & 3 \ 1 & 2 \end{array} \right] \).

Step 3: Change the sign 'b' and 'c'

Change the signs of 'b' (3) and 'c' (1) to get \( \left[ \begin{array}{cc} 4 & -3 \ -1 & 2 \end{array} \right] \).

Step 4: Divide by the determinant

Each term in the matrix \(\left[ \begin{array}{cc} 4 & -3 \ -1 & 2 \end{array} \right] \) is divided by the determinant (5) to find the inverse. The resulting inverse matrix is \( \left[ \begin{array}{cc} 0.8 & -0.6 \ -0.2 & 0.4 \end{array} \right] \).

Key Concepts

Determinant Calculation

In order to understand matrix inversion, it is essential to become familiar with the concept of the determinant. The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant is found using the formula: \( ad - bc \). Here, \(a\), \(b\), \(c\), and \(d\) represent the elements of the matrix in a layout such as:

• \(a\) in the first row, first column,
• \(b\) in the first row, second column,
• \(c\) in the second row, first column,
• \(d\) in the second row, second column.

In practical terms, calculating the determinant quickly tells us whether a matrix is invertible or not. For instance, if the determinant is zero, the matrix does not have an inverse. But if it is non-zero, an inverse exists, and we can proceed to find it.

Matrix Algebra

Matrix algebra involves operations such as addition, subtraction, multiplication, and inversion, which follow certain rules. These rules help us solve systems of linear equations, transformations, and other mathematical functions. Let’s focus on some basic concepts:

• Addition and Subtraction: Matrices must be of the same dimensions to be added or subtracted.
• Multiplication: This operation is more complex, involving the sum of the products of the entries. The number of columns in the first matrix must match the number of rows in the second.
• Identity Matrix: This is a special kind of Square Matrix akin to the number 1 in multiplication. When multiplied by another matrix, it leaves that matrix unchanged.

These foundation concepts in matrix algebra are crucial not only for performing operations like finding the inverse but are integral for higher-level mathematics such as calculus and linear algebra.

Inverse Matrix Calculation

Finding the inverse of a matrix is a fundamental task in various fields including computer science, physics, and engineering. The inverse of a 2x2 matrix can be found using the following steps:To find the inverse of \(\begin{bmatrix}2 & 3 \1 & 4\end{bmatrix}\), start by checking the determinant. As already calculated, the determinant is 5.The steps are as follows:

• Swap the diagonal elements: Exchange positions of the elements on the main diagonal \(a\) and \(d\). This swaps 2 and 4 in our matrix.
• Change the signs of the off-diagonal elements: For elements situated off the main diagonal (\(b\) and \(c\)), change the signs. In our example, change 1 to -1 and 3 to -3.
• Divide Every Element by the Determinant: After forming the new matrix, divide each term by the determinant (5). This gives us the final inverse array.

The final inverse matrix is \(\begin{bmatrix}0.8 & -0.6 \-0.2 & 0.4\end{bmatrix}\). Understanding how to perform these steps accurately ensures you can solve and manipulate complex equations in which the inverse matrix plays a critical role.