Question

Find the adjugate of each of the following matrices. a. \(\left[\begin{array}{rrr}5 & 1 & 3 \\ -1 & 2 & 3 \\ 1 & 4 & 8\end{array}\right]\) b. \(\left[\begin{array}{rrr}1 & -1 & 2 \\ 3 & 1 & 0 \\ 0 & -1 & 1\end{array}\right]\) c. \(\left[\begin{array}{rrr}1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 1\end{array}\right]\) d. \(\frac{1}{3}\left[\begin{array}{rrr}-1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1\end{array}\right]\)

Short answer

Adjugates: a. \( \begin{bmatrix} 4 & 7 & -10 \\ -11 & 33 & 18 \\ 7 & 23 & 11 \end{bmatrix} \), b. \( \begin{bmatrix} 1 & 1 & -1 \\ 0 & 1 & -6 \\ -3 & -3 & 1 \end{bmatrix} \), c. \( \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} \), d. \( \begin{bmatrix} 1 & -\frac{4}{3} & -\frac{4}{3} \\ -\frac{4}{3} & 1 & -\frac{4}{3} \\ -\frac{4}{3} & -\frac{4}{3} & 1 \end{bmatrix} \).

Step-by-step solution

Find the Cofactor Matrix of A for matrix a

To find the cofactor matrix for the given matrix, calculate the determinant of each 2x2 submatrix by removing the current row and column position of each element. Denote the matrix as matrix A:\[A = \begin{bmatrix} 5 & 1 & 3 \ -1 & 2 & 3 \ 1 & 4 & 8 \end{bmatrix}\]For example, the (1,1) entry's cofactor can be calculated as the determinant of the matrix formed by deleting the first row and first column:\[ \begin{vmatrix} 2 & 3 \ 4 & 8 \end{vmatrix} = (2)(8) - (3)(4) = 16 - 12 = 4 \]Repeat this procedure to fill in the cofactor matrix for all elements.

Apply Alternating Sign Pattern for matrix a

After calculating all the determinants, apply the alternating sign pattern to get the cofactor matrix:\[Cof(A) = \begin{bmatrix} +4 & -(11) & +(7) \ -(-7) & +(33) & -(23) \ +(-10) & -(-18) & +(11) \end{bmatrix} = \begin{bmatrix} 4 & -11 & 7 \ 7 & 33 & 23 \ -10 & 18 & 11 \end{bmatrix} \]

Transpose the Cofactor Matrix for matrix a

Transpose the cofactor matrix to find the adjugate matrix. The transposed cofactor matrix becomes:\[Adj(A) = \begin{bmatrix} 4 & 7 & -10 \-11 & 33 & 18 \ 7 & 23 & 11 \end{bmatrix}\]

Repeat Steps 1-3 for matrix b

Let's denote the second matrix as matrix B:\[B = \begin{bmatrix} 1 & -1 & 2 \ 3 & 1 & 0 \ 0 & -1 & 1 \end{bmatrix}\]Calculate the cofactor for each element of B using the same method from Step 1, apply the alternating sign from Step 2, and then transpose to get the adjugate matrix.

Find the Cohactors of Matrix B

Cofactors of matrix B are calculated similarly:\[Cof(B) = \begin{bmatrix} +1 & 0 & -3 \ +1 & 1 & -3 \ -1 & -6 & 1 \end{bmatrix} \]

Adjugate of Matrix B

Transpose of the cofactor matrix:\[Adj(B) = \begin{bmatrix} 1 & 1 & -1 \ 0 & 1 & -6 \ -3 & -3 & 1 \end{bmatrix} \]

Repeat Steps 1-3 for matrix c

Repeat the process of finding cofactors and forming the adjugate matrix for matrix c:\[C = \begin{bmatrix} 1 & 0 & -1 \ -1 & 1 & 0 \ 0 & -1 & 1 \end{bmatrix}\]

Find the Cofactors of Matrix C

Cofactors of matrix C:\[Cof(C) = \begin{bmatrix} +1 & 0 & -1 \ 0 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix} \]

Adjugate of Matrix C

Transpose the cofactor matrix:\[Adj(C) = \begin{bmatrix} 1 & 0 & -1 \ 0 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix} \]

Repeat Steps 1-3 for matrix d

Note that matrix D has a scalar factor \(\frac{1}{3}\) which can be ignored for the calculation of cofactors. Denote the matrix as matrix D without the scalar:\[D = \begin{bmatrix} -1 & 2 & 2 \ 2 & -1 & 2 \ 2 & 2 & -1 \end{bmatrix}\]Calculate the cofactor for each element, apply the alternating sign and transpose.

Find the Cofactors of Matrix D

Calculate cofactor matrix for D:\[Cof(D) = \begin{bmatrix} 3 & - 4 & - 4 \ - 4 & 3 & - 4 \ - 4 & - 4 & 3 \end{bmatrix}\]

Adjugate of Matrix D

Transpose the cofactor matrix of unscaled D:\[Adj(D) = \begin{bmatrix} 3 & -4 & -4 \ -4 & 3 & -4 \ -4 & -4 & 3 \end{bmatrix} \]Multiply by the original scalar \(\frac{1}{3}\) from matrix D:\[Adj \left( \frac{1}{3}D \right) = \begin{bmatrix} 1 & -\frac{4}{3} & -\frac{4}{3} \ -\frac{4}{3} & 1 & -\frac{4}{3} \ -\frac{4}{3} & -\frac{4}{3} & 1 \end{bmatrix} \]

Key concepts

Matrix Theory

Matrix theory is a fundamental part of linear algebra that deals with mathematical structures called matrices. A matrix is essentially an array of numbers or functions arranged in a rectangular format with rows and columns.
This concept is widely used to solve systems of linear equations and to represent linear transformations. Matrices come in various types, such as square, rectangular, diagonal, and identity matrices, each serving different purposes in complex calculations.
In the context of linear transformations, a matrix can be seen as a way to describe the change of coordinates, making them an essential tool in graphical applications and engineering.

• Square Matrices: These have the same number of rows and columns.
• Diagonal Matrices: All non-diagonal elements are zero.
• Identity Matrices: Diagonal elements are all one, and all others are zero, serving as the multiplicative identity in matrix operations.

Understanding the different types of matrices and their properties is crucial for manipulating and solving matrix equations.

Cofactor Matrix

The cofactor matrix is derived from the original matrix by calculating the cofactor of each element. To find the cofactor of an element in a matrix, you need to follow these steps:
1. Remove the row and column where the element is located to form a new smaller matrix.
2. Compute the determinant of this smaller matrix.
3. Apply an alternating plus and minus sign pattern ( (-1)^{i+j} ) based on the element's position (i,j).
This process gives each element a cofactor, which collectively forms the cofactor matrix. The cofactor matrix is especially important when computing the adjugate matrix, which is the transpose of the cofactor matrix.

• Step-by-Step: Calculate determinants, apply signs, and produce the cofactor matrix.
• Sign Pattern: Helps ensure accuracy in calculations.

Calculating cofactors is fundamental when dealing with determinant calculations and finding inverses of matrices.

Determinant Calculation

Calculating the determinant is a crucial operation in matrix algebra, particularly for square matrices. The determinant is a scalar value that captures important properties of the matrix, such as invertibility and linear dependency of rows or columns.
For a 2x2 matrix, the determinant is straightforward: \[ \text{det}( \begin{bmatrix} a & b \ c & d \end{bmatrix} ) = ad - bc. \] For larger matrices, determinants are calculated using cofactor expansion, which requires reducing the matrix to smaller 2x2 matrices.
This reduction involves choosing any row or column and summing the products of its elements by their corresponding cofactors. The determinant is also closely tied to the volume scaling of geometric shapes when matrices represent transformations.

• Properties: Zero determinant indicates non-invertibility.
• Applications: Used to solve systems of linear equations via Cramer's Rule.

Understanding how to compute and interpret the determinant is essential for advanced matrix operations and solving linear systems.

Linear Algebra

Linear algebra is the branch of mathematics that studies vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations. At its core, it provides the theoretical framework for solving linear systems and for understanding the properties and relationships of matrices and vectors.
Matrices are used to represent linear transformations, which are functions that map vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication.
Key concepts in linear algebra include vector addition, scalar multiplication, span, basis, and dimension of vector spaces. Linear algebra is widely applied in various fields, including computer science, physics, engineering, and economics.

• Vector Spaces: Fundamental building blocks characterized by dimensions.
• Linear Transformations: Functions that map vectors while preserving linear structure.

Mastering linear algebra is indispensable for understanding modern applications in data science, machine learning, and computer graphics.