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Magazine Chapter 6: Problem 28
For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 0.8 & 0.2 & 0.1 \\ -0.2 & 0 & 0.3 \\ 0 & 0 & 0.5 \end{array}\right]$$
Short Answer
Expert verified
The inverse of matrix \( A \) is \( \begin{pmatrix} 0 & 7.5 & 0 \ 0 & 5 & 0 \ 0 & -8 & 2 \end{pmatrix} \).
Step by step solution
01
Check if the matrix is square
The given matrix \( A \) is a 3x3 matrix, which means it is square. Only square matrices can have an inverse.
02
Compute the determinant of the matrix
The determinant \( \text{det}(A) \) of the matrix \( A \) needs to be non-zero for the inverse to exist. Calculate the determinant using the formula for a 3x3 matrix:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]Substitute the values from matrix \( A \):\[\text{det}(A) = 0.8(0 \times 0.5 - 0 \times 0.3) - 0.2(-0.2 \times 0.5 - 0 \times 0.1) + 0.1(-0.2 \times 0 - 0 \times 0.3)\]\[= 0.8(0) - 0.2(0.1) + 0.1(0)\]\[= 0.02\]
03
Determine if the inverse exists
Since the determinant of \( A \) is \( 0.02 \), which is not zero, the inverse of \( A \) exists.
04
Compute the adjugate matrix
Find the adjugate (adjoint) of matrix \( A \) which is the transpose of the cofactor matrix.The cofactor matrix of \( A \) can be calculated by finding the determinant of each 2x2 minor and applying the checkerboard rule of signs. Using matrix \( A \), this will be:\[\operatorname{cofactor}(A) = \begin{pmatrix} 0 & 0 & 0 \ 0.15 & 0.1 & -0.16 \ 0 & 0 & 0.04 \end{pmatrix}\]Transpose this matrix to get the adjugate:\[\text{adj}(A) = \begin{pmatrix} 0 & 0.15 & 0 \ 0 & 0.1 & 0 \ 0 & -0.16 & 0.04 \end{pmatrix}\]
05
Calculate the inverse of the matrix
The inverse of matrix \( A \) is calculated using the formula:\[A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)\]Substitute \( \text{det}(A) = 0.02 \) and \( \text{adj}(A) \) into the formula:\[A^{-1} = \frac{1}{0.02} \times \begin{pmatrix} 0 & 0.15 & 0 \ 0 & 0.1 & 0 \ 0 & -0.16 & 0.04 \end{pmatrix}\]\[= \begin{pmatrix} 0 & 7.5 & 0 \ 0 & 5 & 0 \ 0 & -8 & 2 \end{pmatrix}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant Calculation
The determinant is a crucial concept when determining if a square matrix has an inverse. For a 3x3 matrix, like our example matrix \( A \), the determinant is calculated using the formula:
- For the matrix elements \( a, b, c, d, e, f, g, h, i \), the formula is \( det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
- Assign the elements: \( a = 0.8, b = 0.2, c = 0.1, d = -0.2, e = 0, f = 0.3, g = 0, h = 0, i = 0.5 \).
- Plug these into the formula: \( det(A) = 0.8(0 \times 0.5 - 0 \times 0.3) - 0.2(-0.2 \times 0.5 - 0 \times 0.1) + 0.1(-0.2 \times 0 - 0 \times 0.3) \).
- Compute the result: \( 0.8(0) - 0.2(0.1) + 0.1(0) = 0.02 \).
Adjugate Matrix
The adjugate matrix is essential in finding the inverse of a square matrix. It is formed by transposing the cofactor matrix, a process that involves minor matrices and signed cofactors.
- The cofactor matrix is created by taking each element, finding its minor (the determinant of the matrix formed by deleting its row and column), and applying signs based on a checkerboard pattern of plus and minus.
- In our example: minor matrices are evaluated, and signs are applied.
- This results in the cofactor matrix, which for matrix \( A \) is \( \begin{pmatrix} 0 & 0 & 0 \ 0.15 & 0.1 & -0.16 \ 0 & 0 & 0.04 \end{pmatrix} \).
- The adjugate matrix is simply the transpose of this cofactors matrix, giving us \( \begin{pmatrix} 0 & 0.15 & 0 \ 0 & 0.1 & 0 \ 0 & -0.16 & 0.04 \end{pmatrix} \).
Cofactor Matrix
Cofactors play a significant role in matrix algebra, particularly in finding the adjugate and the determinant.
- Each element of a matrix has a corresponding cofactor, which is crucial for both the adjunct and determinant calculations.
- A cofactor is determined by excluding the row and column of an element to find the minor (a smaller matrix) and calculating its determinant.
- Then, multiply this determinant by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.
- For instance, the element \( b_{22} = 0.3 \) has a minor matrix \( \begin{pmatrix} 0.8 & 0.2 \ 0 & 0 \end{pmatrix} \) with a determinant of \(-0.16\). Applying the checkerboard rule, the cofactor becomes \(-0.16\).
- Perform this for each element, leading to the cofactor matrix \( \begin{pmatrix} 0 & 0 & 0 \ 0.15 & 0.1 & -0.16 \ 0 & 0 & 0.04 \end{pmatrix} \).
Square Matrices
Square matrices are matrices with the same number of rows and columns, and they are foundational in matrix algebra.
- They are required to have an inverse because only square matrices allow the calculation of determinants, adjuncts, and inverses.
- For example, our matrix \( A \) is a 3x3 square matrix and thus qualifies for inverse calculations.
- An inverse exists only if the determinant is non-zero, as demonstrated with matrix \( A \).
- This quality of square matrices makes them particularly interesting and widely used in solutions involving system of equations, transformations, and more.
