Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rrr} 1 & -3 & -2 \\ -1 & 3 & 1 \\ 0 & 2 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of matrix A is 6 and the determinant of its inverse is 1/6. This indicates that the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.

Step 1: Find the determinant of Matrix A

The determinant of a matrix can be calculated in different ways but the most straightforward for this exercise is to use the rule of Sarrus. Expanding by minors, we get \( \text{det}(A) = (1)(3)(-2) - (-3)(-1)(-2) - (-2)(3)(0) - (0)(3)(1) - (1)(2)(-1) - (-3)(-1)(2) = 6.

Step 2: Find the inverse of Matrix A

The inverse of a matrix can be found by following the formula \(A^{-1} = \frac{1}{\text{det}(A)} adj(A)\) where adj(A) is the adjugate of A. Adj(A) can be calculated by transposing the matrix of cofactors C of A. Cofactors can be found by taking each element of A, removing the row and column that element is part of and calculating the determinant of the remaining 2x2 matrix, and alternating the signs for each element starting with a positive sign in the top left (+ - + on the first row, - + - on the second and so on). After calculating, we get \(A^{-1} = \frac{1}{6} \times \left[\begin{array}{ccc} -8 & -2 & -2 \ -2 & -1 & 1 \ -6 & -1 & -1 \end{array}\right]\)

Step 3: Find the determinant of \(A^{-1}\)

The determinant of \(A^{-1}\) can be calculated in the same manner as in step 1. After applying the rule of Sarrus again, we get \( \text{det}(A^{-1}) = \frac{1}{6}\), which is precisely the inverse of the determinant of the orginal matrix A.

Step 4: Compare the results and conclude

Comparing the results from part (a) and part (c), the determinant of the inverse of matrix A is the reciprocal of the determinant of matrix A. This makes sense mathematically as the inverse of a matrix should 'undo' the transformation that A applies, so their determinants should be reciprocals.

Key Concepts

Matrix Inverse

Finding the inverse of a matrix is a fundamental task in linear algebra. A matrix inverse, denoted as \( A^{-1} \), is a matrix that when multiplied by the original matrix \( A \) yields the identity matrix, often symbolized as \( I \). The identity matrix has ones on the diagonal and zeros elsewhere, functioning like the number '1' for matrix multiplication.
To calculate the matrix inverse, we use the formula:

• \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \)

The determinant of \( A \), \( \text{det}(A) \), must be non-zero for the inverse to exist because dividing by zero is undefined. The adjugate matrix, adj(A), plays a crucial role here, and we will cover more about it soon. Importantly, each step in working with inverses aims at achieving an operational reverse to the transformations that the matrix \( A \) performs.

Adjugate Matrix

The adjugate of a matrix, sometimes called the adjoint, is the next piece needed after you have found the determinant when calculating a matrix inverse. The adjugate matrix is found by following a systematic method:

• First, compute the cofactor matrix by making each element into a minor determinant and applying a checkerboard pattern of signs (+ and -).
• This involves removing the row and column of each element of the matrix and calculating the determinant of the resulting 2x2 sub-matrix.
• Transpose this cofactor matrix. This transposing swaps rows with columns across the diagonal.

For example, if you have matrix \( A \), its adjugate matrix will be computed by rearranging the cofactors into a new layout. This step is crucially important as it directly feeds into finding the inverse of \( A \). The adjugate essentially "adjusts" these minor calculations to fit into the bigger matrix picture. Because %correct adjugate movement is vital to obtaining a correct inverse.

Determinant Properties

Determinants have several interesting properties that have practical implications in linear algebra. A matrix determinant is a scalar value that can indicate whether a matrix is invertible. Specifically, if \( \text{det}(A) eq 0 \), then \( A \) has an inverse, otherwise, it does not.
A fascinating principle regarding determinants is that the determinant of the inverse matrix \( A^{-1} \) is the reciprocal of \( \text{det}(A) \):

• \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \)

This makes sense because the operation of inversion in matrix algebra is akin to flipping something over, just as taking the reciprocal of a number gives it a representative inversion in scalar terms. Determinants also help to simplify complex operations such as finding eigenvalues or solving systems of equations, making them indispensable in various applications.