Problem 76 - 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. $$A=\left[\begin{array}{rrr} -1 & 3 & 2 \\ 1 & 3 & -1 \\ 1 & 1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of A is -5. The inverse of A is \[ A^{-1} = \left[\begin{array}{rrr}-1 & -1 & 3 \ 1 & -1 & 1 \ 1 & 1 & -3\end{array}\right] \] . The determinant of the inverse of A is -1/5. From these results, we can infer that the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix, i.e., \(\text{det}(A^{-1}) = 1/(\text{det}(A))\).

Step 1: Calculate the Determinant of A

The determinant for a 3x3 matrix \( A = \left[\begin{array}{rrr}-1 & 3 & 2 \ 1 & 3 & -1 \ 1 & 1 & -2\end{array}\right] \) can be calculated using the formula: \[\text{det}(A) = a(ei−fh) - b(di−fg) + c(dh−eg) \]Using the values of A, we get det(A) = -1((3*-2) - (-1*1)) - 3((1*-2) - (-1*1)) + 2((1*1) - (3*1)) = -5.

Step 2: Calculate the Inverse of A

The inverse of a 3x3 matrix can be found using the formula \[A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)\]where adj(A) is the adjugate of A. The adjugate of A can be found by taking the transpose of the cofactor matrix of A. After the calculations, we get \[A^{-1} = \left[\begin{array}{rrr}-1 & -1 & 3 \ 1 & -1 & 1 \ 1 & 1 & -3\end{array}\right]\]

Step 3: Find the Determinant of the Inverse of A

Now we calculate the determinant of \( A^{-1} \). Repeating the determinant process from Step 1 yields a determinant of -1/5.

Step 4: Compare the results and make a conjecture

From the results in steps 1 and 3, we observe that the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix. Therefore, the conjecture is: \(\text{det}(A^{-1}) = 1/(\text{det}(A))\).

Key Concepts

Inverse Matrix

An inverse matrix is essentially the reverse of the original matrix, often denoted as \(A^{-1}\). If a matrix \(A\) is invertible, multiplying it by its inverse, \(A \times A^{-1}\), will result in the identity matrix, similar to multiplying a number by its reciprocal to get 1.
This is fundamental because the identity matrix acts like the number 1 in matrix operations.
To find \(A^{-1}\), we use the formula:

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

Here, \(\text{adj}(A)\) is the adjugate of \(A\), and \(\text{det}(A)\) is the determinant of \(A\).
Keep in mind, an inverse matrix only exists for square matrices with non-zero determinants. This means the determinant must be calculated first to ensure \(A\) is invertible.
For the given matrix, the inverse was successfully calculated since the determinant \(\text{det}(A) = -5\), which is non-zero.

Adjugate Matrix

The adjugate matrix, also known as adjoint matrix, plays a crucial role in finding the inverse of a matrix.
It is a matrix where each element is replaced by its corresponding cofactor, and then the entire matrix is transposed.
To find the adjugate matrix:

• Compute the cofactor matrix of \(A\).
• Transpose the cofactor matrix to form the adjugate.

The adjugate is used in calculating the inverse as follows:

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

Thus, understanding the adjugate is key for matrix inversion.
Transposing the cofactor matrix ensures that every element is positioned correctly in the inverse calculation.

Cofactor Matrix

The cofactor matrix is built from the original matrix \(A\) and includes calculating minors and cofactors.
Here's how it is constructed:

• Choose a specific element in the matrix.
• Eliminate its row and column.
• Calculate the determinant of the resulting smaller matrix (this is the minor).
• Obtain the cofactor by applying a checkerboard pattern of positive and negative signs to the minor.

By arranging all cofactors into a matrix, you create the cofactor matrix.
Once the cofactor matrix is calculated, it gets transposed to form the adjugate matrix.
This is a critical step in deriving the matrix inverse.

3x3 Matrix

Understanding a 3x3 matrix is crucial as it's a common type of square matrix with applications across mathematics and engineering.
A 3x3 matrix, like other square matrices, can have a determinant, inverse, and adjugate.
Such matrices are often written in the form:

• \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]

Identifying determinants of 3x3 matrices is a common task. The formula involves a combination of matrix elements and helps detect whether the matrix is invertible.
These matrices require systematic approaches for computations, including methods like row reduction or leveraging calculators for complex operations.
Their application ranges from computer graphics to physics, making them essential for students to grasp thoroughly.