Problem 16 - Precalculus with Limits

Question

In Exercises 13-24, find the inverse of the matrix (if it exists). \(\left[ \begin{array}{r} -7 & 33 \\ 4 & -19 \end{array} \right]\)

Step-by-Step Solution

Answer

\(M^{-1} = \left[ \begin{array}{cc} 19/133 & 33/133 \ 4/133 & 7/133 \end{array} \right]\)

Step 1: Calculate the Determinant

For a 2x2 matrix \(M = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), the determinant is calculated as \(Det(M) = (ad - bc)\). Here \(a = -7\), \(b = 33\), \(c = 4\), and \(d = -19\). Therefore, \(Det(M) = ((-7)*(-19))-((33)*(4)) = 133.\)

Step 2: Swapping and changing signs

Swap the first \(a = -7\) and the last \(d = -19\) elements: \(a',d' = d,a = -19,-7\). Then, change the signs of the other two elements \(b' = -b = -33\) and \(c' = -c = -4\)

Step 3: Find the inverse of the matrix

If \(M'\) is the matrix obtained after performing the operations in the previous step and \(Det(M) ≠ 0\), then the inverse of the matrix \(M\), denoted as \(M^{-1}\), is obtained as follows \(M^{-1} = (1/Det(M))*M'\). Substituting the values calculated previously results in, \(M^{-1} = (1/133)*\left[ \begin{array}{cc} -19 & -33 \ -4 & -7 \end{array} \right]\), Simplifying this gives us \(M^{-1} = \left[ \begin{array}{cc} 19/133 & 33/133 \ 4/133 & 7/133 \end{array} \right]\).

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the formula for the determinant is very straightforward: if you have a matrix \(M\) represented as \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), the determinant is \(Det(M) = ad - bc\).
This concept is crucial because the determinant tells us if a matrix is invertible. If the determinant is zero, then the matrix doesn't have an inverse. In the given exercise, the matrix \(\left[ \begin{array}{r} -7 & 33 \ 4 & -19 \end{array} \right]\) has a determinant of \(133\), calculated as follows:

• Multiply \(a\) and \(d\): \((-7) \times (-19) = 133\)
• Multiply \(b\) and \(c\): \(33 \times 4 = 132\)
• Subtract the second product from the first: \(133 - 132 = 1\)

Therefore, the determinant is not zero, indicating that an inverse exists.

Inverse Matrix

Finding the inverse of a matrix is akin to reversing a mathematical operation. While numbers have reciprocals, matrices have inverses. To calculate an inverse for a 2x2 matrix, you need the determinant. An inverse exists only when the determinant is non-zero.
For a matrix \(M = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), the process involves a few steps:

• Calculate the determinant \(ad - bc\).
• Swap the positions of \(a\) and \(d\).
• Change the signs of \(b\) and \(c\).
• Multiply by \(1/Det(M)\).

For example, when calculating the inverse of \(\left[ \begin{array}{r} -7 & 33 \ 4 & -19 \end{array} \right]\), after adjusting the matrix to \(\left[ \begin{array}{cc} -19 & -33 \ -4 & -7 \end{array} \right]\) and confirming the determinant is \(133\), you multiply this matrix by \(1/133\) to find the inverse, leading to a practical inverse that is easy to work with in various calculations.

2x2 Matrix Calculation

Matrix calculations can initially seem daunting, but 2x2 matrices are a perfect starting point due to their simplicity. Operations with these matrices involve mathematical rules such as addition, multiplication, and finding inverses, each relying on the arrangement of two rows and two columns.
In this exercise, we are concerned with inversion. This involves using specific steps like determinant calculation and element swapping, which limit complexity while still allowing for meaningful operations.

• With a given matrix \(\left[ \begin{array}{r} -7 & 33 \ 4 & -19 \end{array} \right]\), every step focuses on these elements.
• Numerical outputs are clean and manageable, as seen with resulting fractions like \(19/133\).

These calculations underline the beauty of matrix algebra in solving linear equations and transforming geometric problems. Each number and operation has a purpose, reflecting a balance between abstract concepts and tangible results.