Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rr} 0 & 4 \\ -3 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix exists and is `[0.5 -0.33; 0.25 0]`.

Step 1: Determining the Determinant

The determinant of a 2x2 matrix [a b; c d] is given by ad - bc. Thus, the determinant here will be \((0 * 6) - (4 * -3) = 12\).

Step 2: Check for existence of inverse

If the determinant is zero, the inverse of a matrix does not exist. In this case, the determinant is 12, which is not zero. Therefore, the inverse of this matrix does exist.

Step 3: Calculate the inverse

The formula for the inverse of a 2x2 matrix [a b; c d] is (1/determinant) * [d -b; -c a]. Using this formula here, the inverse of the given matrix is equal to 1/12 * `[6 -4; 3 0]` = `[0.5 -0.33; 0.25 0]`.

Key Concepts

Determinants

In linear algebra, the determinant is a special number calculated from a square matrix. It provides crucial information about the matrix, including whether it can be inverted. For a 2x2 matrix, the determinant is computed using the formula: \\[\text{{det}}(A) = ad - bc\]\where \(a, b, c,\) and \(d\) are the elements of the matrix. To find the determinant of the given matrix \\[\begin{bmatrix} 0 & 4 \ -3 & 6 \end{bmatrix}\]\we use these elements in the formula. Substituting them into the formula, we get \\(0 \times 6 - 4 \times (-3) = 12\). As 12 is not zero, it indicates that the matrix is invertible. Thus, knowing how to calculate the determinant is key to determining if a 2x2 matrix has an inverse.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix with two rows and two columns. It is often represented in the form \\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]\where \(a, b, c,\) and \(d\) are the elements of the matrix. These matrices are commonly used in various mathematical applications. For instance, they can represent transformations in 2D space, like rotations and scalings. The operations with such matrices often involve basic algebraic rules, making them an excellent starting point for understanding more complex matrix operations, such as determining the inverse, calculating eigenvalues, and performing matrix multiplication.

Inverse Matrix Calculation

Finding the inverse of a matrix is essentially about finding another matrix which, when multiplied with the original, yields the identity matrix. For a 2x2 matrix \\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]\the inverse, if it exists, is given by \\[\frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\].\This process requires that the determinant \((ad - bc)\) is not zero. For the given matrix \\[\begin{bmatrix} 0 & 4 \ -3 & 6 \end{bmatrix}\],\we already calculated the determinant as 12, which isn't zero, indicating an inverse exists. Applying the inverse formula, we substitute the values to get \\[\frac{1}{12} \begin{bmatrix} 6 & -4 \ 3 & 0 \end{bmatrix} = \begin{bmatrix} 0.5 & -0.33 \ 0.25 & 0 \end{bmatrix}\].\This derived inverse matrix is useful in a variety of contexts, such as solving systems of linear equations and understanding linear transformations.