13.2 Matrix operations and determinants

Matrices are the backbone of linear algebra, allowing us to represent and manipulate complex systems of equations. In this section, we'll learn how to perform operations like addition, multiplication, and finding inverses of matrices.

Understanding matrix operations is crucial for solving real-world problems in physics, engineering, and computer science. We'll explore determinants, a powerful tool for analyzing matrices, and discover how they relate to matrix properties and transformations.

Matrix Operations

Adding and Multiplying Matrices

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• Matrix addition performed by adding corresponding elements of two matrices with the same dimensions
• Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products
  • The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible
  • The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix
• Transpose of a matrix $A$, denoted as $A^T$, is obtained by interchanging the rows and columns of the original matrix
  • For an $m \times n$ matrix, the transpose is an $n \times m$ matrix
• Trace of a square matrix is the sum of the elements on the main diagonal (from top left to bottom right)
  • For an $n \times n$ matrix $A$, the trace is given by $tr(A) = a_{11} + a_{22} + \ldots + a_{nn}$

Properties of Matrix Operations

• Matrix addition is commutative: $A + B = B + A$
• Matrix addition is associative: $(A + B) + C = A + (B + C)$
• Matrix multiplication is associative: $(AB)C = A(BC)$
• Matrix multiplication is distributive over addition: $A(B + C) = AB + AC$
• Matrix multiplication is not commutative in general: $AB \neq BA$
• The product of a matrix and its transpose is always a symmetric matrix: $(AB)^T = B^T A^T$

Matrix Inversion

Inverse and Adjugate Matrices

• The inverse of a square matrix $A$, denoted as $A^{-1}$, is a matrix such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix
  • Not all matrices have an inverse; a matrix with an inverse is called invertible or non-singular
• To find the inverse of a matrix, first calculate the determinant (denoted as $[det(A)](https://www.fiveableKeyTerm:det(a))$ or $[|A|](https://www.fiveableKeyTerm:|a|)$)
  • If the determinant is non-zero, the matrix is invertible
• The adjugate matrix (also called the adjoint) is the transpose of the cofactor matrix
• The cofactor matrix is obtained by replacing each element of the original matrix with its cofactor
  • The cofactor of an element $a_{ij}$ is $(-1)^{i+j}$ times the determinant of the submatrix formed by deleting the $i$-th row and $j$-th column

Calculating the Inverse Matrix

• The inverse of a matrix $A$ can be calculated using the formula: $A^{-1} = \frac{1}{det(A)} adj(A)$, where $adj(A)$ is the adjugate matrix
• For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the inverse is given by: $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
• For larger matrices, the process involves calculating the cofactor matrix, finding the adjugate matrix, and then dividing by the determinant

Matrix Properties

Determinants and Their Properties

• The determinant is a scalar value associated with a square matrix that provides information about the matrix's properties
• For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $det(A) = ad - bc$
• For larger matrices, the determinant can be calculated using cofactor expansion along any row or column
• Properties of determinants:
  • The determinant of the identity matrix is always 1
  • If two rows or columns of a matrix are interchanged, the determinant changes sign
  • If a matrix has a row or column of zeros, its determinant is zero
  • The determinant of a triangular matrix (upper or lower) is the product of its diagonal entries

Rank of a Matrix

• The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix
• A set of vectors is linearly independent if no vector can be expressed as a linear combination of the others
• The rank of a matrix is always less than or equal to the smaller dimension of the matrix
  • For an $m \times n$ matrix, the rank is at most $min(m, n)$
• The rank of a matrix can be determined by performing Gaussian elimination and counting the number of non-zero rows in the reduced row echelon form
• A matrix is said to have full rank if its rank equals the smaller dimension of the matrix