Matrix Properties and Remembering Tricks

December 16, 2024December 16, 2024by Kurious Fox

1. Basic Additive Properties of Matrices:

Commutative Property of Addition:
$A + B = B + A$
Associative Property of Addition:
$(A + B) + C = A + (B + C)$
Additive Identity:
$A + 0 = A$
(Where $0$ is the zero matrix.)
Additive Inverse:
$A + (-A) = 0$

Ways to remember: We see that the basic properties of matrix addition are quite similar to those of real numbers, with the commutative property, associative property, and the existence of an identity element.

2. Properties of Matrix Multiplication:

Distributive Property:
$A(B + C) = AB + AC \quad \text{and} \quad (A + B)C = AC + BC$
Associative Property:
$A(BC) = (AB)C$
Non-Commutativity:
$AB \neq BA \quad \text{(in general).}$
Identity Matrix ( $I$ ):
$AI = IA = A$

Memory tip: Thus, we see that the basic properties of matrix multiplication are quite similar to those of real numbers, except that matrix multiplication is not commutative.

3. Properties of Transpose ( $A^T$ ):

Transpose of a Sum:
$(A + B)^T = A^T + B^T$
Transpose of a Product:
$(AB)^T = B^T A^T$
Double Transpose:
$(A^T)^T = A$
Symmetric Matrices:
If $A$ is symmetric ( $A^T = A$ ), then $A$ is called a symmetric matrix.

4. Special Matrices and Their Properties:

Square Matrix: Number of rows = number of columns.
Triangular Matrices:
Upper triangular matrix: Elements below the main diagonal are zero.
Lower triangular matrix: Elements above the main diagonal are zero.
Identity Matrix ( $I$ ): Main diagonal is all 1s, and other elements are 0.
Orthogonal Matrix ( $Q$ ):
$Q^T Q = Q Q^T = I$
Inverse Matrix ( $A^{-1}$ ):
$A A^{-1} = A^{-1} A = I$

5. Determinants and Rank of a Matrix:

Determinant of a Product:
$\det(AB) = \det(A) \cdot \det(B)$
Determinant of a Transpose:
$\det(A^T) = \det(A)$
Singular Matrix: If $\det(A) = 0$ , then $A$ is non-invertible.
Rank of a Matrix: The number of linearly independent rows or columns.

6. Properties Related to Eigenvalues and Eigenvectors:

Matrix $A$ and its Transpose $A^T$ have the same eigenvalues.
Product of eigenvalues = $\det(A)$ .
Sum of eigenvalues = the sum of the elements on the main diagonal.

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Matrix Properties and Remembering Tricks