Problem 9 - Precalculus: Mathematical for Calculus

Question

Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}2 & 5 \\ -5 & -13\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix is \(\begin{pmatrix} 13 & 5 \\ -5 & -2 \end{pmatrix}\).

Step 1: Check if the Matrix is Invertible

To determine if the matrix is invertible, we must check its determinant. The given matrix is \(A = \begin{pmatrix} 2 & 5 \ -5 & -13 \end{pmatrix}\). The determinant \(\text{det}(A)\) for a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is calculated as \(ad - bc\). Here, \(a = 2\), \(b = 5\), \(c = -5\), and \(d = -13\). Substitute these into the formula to find the determinant: \(2(-13) - 5(-5) = -26 + 25 = -1\). Since the determinant is not zero, the matrix is invertible.

Step 2: Compute the Inverse of the Matrix

Once confirmed that the matrix is invertible, calculate the inverse using the formula for a 2x2 matrix, \(A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\). With \( \text{det}(A) = -1 \), calculate the inverse: \(-1 \times \begin{pmatrix} -13 & -5 \ 5 & 2 \end{pmatrix} = \begin{pmatrix} 13 & 5 \ -5 & -2 \end{pmatrix}\). Thus, \( A^{-1} = \begin{pmatrix} 13 & 5 \ -5 & -2 \end{pmatrix} \).

Key Concepts

Determinant Calculation

The determinant of a matrix is a special value that can tell us whether a matrix has an inverse or not. For a 2x2 matrix like the one provided:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]its determinant is calculated using the formula \( ad - bc \). This means you multiply \( a \) by \( d \), then subtract the product of \( b \) and \( c \). In the given exercise, the matrix is:\[\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\]The values for \( a, b, c, \) and \( d \) are 2, 5, -5, and -13, respectively. Calculate the determinant:* Multiply \( 2 \) by \(-13\) to get \(-26\).* Multiply \( 5 \) by \(-5\) to get \(-25\).* Subtract these results: \(-26 + 25 = -1\).If the determinant is not zero, the matrix can potentially have an inverse, as seen here since \(-1\) is not zero.

2x2 Matrix Inversion

After verifying a non-zero determinant, the next step is finding the inverse of the matrix. The formula for the inverse of a 2x2 matrix \(A\) is:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Here, the determinant \(\text{det}(A)\) is -1. Substitute the original elements of the matrix \(a = 2\), \(b = 5\), \(c = -5\), and \(d = -13\) into the formula:* Replace \(d\) with -13, \(-b\) with -5, \(-c\) with 5, and \(a\) with 2.* Calculate the inverse by distributing the \(-1\): * Multiply each element in the new matrix by \(-1\):\[\begin{bmatrix}-13 & -5 \ 5 & 2\end{bmatrix} \to \begin{bmatrix}13 & 5 \ -5 & -2\end{bmatrix}\]Therefore, the inverse matrix \(A^{-1}\) is:\[\begin{bmatrix}13 & 5 \ -5 & -2\end{bmatrix}\]

Invertible Matrix Conditions

Understanding when a matrix is invertible is crucial. A matrix is invertible if and only if its determinant is non-zero. This is a key condition because a zero determinant means the matrix cannot be undone or reversed. When dealing with 2x2 matrices, follow these steps:

• Calculate the determinant using \(ad - bc\).
• If the determinant is 0, the matrix is non-invertible, meaning it doesn't have an inverse.
• If the determinant is non-zero, proceed to find its inverse using the inversion formula.

This process ensures a matrix can be part of solving systems of linear equations or in applications of computer graphics, where matrices often need to be manipulated and reversed.