Matrices

This chapter systematically studies the basic concepts, operations, properties, elementary transformations, inverse matrices, adjugate matrices, and rank of matrices.

Definition and Types of Matrices

• $m \times n$ matrix: $A = (a_{ij})_{m \times n}$
• Square matrix, zero matrix, identity matrix, diagonal matrix, symmetric matrix, skew-symmetric matrix

Matrix Operations

• Addition, scalar multiplication: add corresponding elements of matrices of the same type, scalar multiplication multiplies each element
• Multiplication: $A_{m \times n} B_{n \times p}$, $C_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$
• Transpose: $A^T$

Properties of Matrices

• Addition and scalar multiplication satisfy commutative, associative, and distributive laws
• Multiplication satisfies associative and distributive laws, but not commutative law
• $(AB)^T = B^T A^T$
• $(A^T)^T = A$

Inverse and Adjugate Matrices

• Inverse matrix: $AA^{-1} = A^{-1}A = I$
• Invertibility condition: $|A| \neq 0$
• Adjugate matrix: $A^* = (A_{ji})$, $A^{-1} = \frac{1}{|A|}A^*$

Elementary Transformations and Elementary Matrices

• Elementary row (column) transformations: swap, add multiple, scalar multiplication
• Elementary matrix: obtained by performing an elementary transformation on the identity matrix
• One-to-one correspondence between elementary matrices and elementary transformations

Rank and Equivalence of Matrices

• Rank: the maximum number of linearly independent rows (or columns)
• Equivalence: can be transformed into each other by a finite number of elementary transformations
• Block matrices and their operations

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Exercises

1. Determine the types of the following matrices: $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}$.
2. Compute $A + B$, where $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$.
3. Compute $AB$, $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$.
4. Determine whether $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is invertible. If so, find $A^{-1}$.
5. Find the adjugate matrix of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.
6. Write the elementary matrix that swaps the first and second rows of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.
7. Find the rank of $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{pmatrix}$.

Reference Answers

1. Determine the types of the following matrices

$A$: $2 \times 2$ identity matrix, symmetric matrix; $B$: $2 \times 2$ symmetric matrix

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2. Compute $A + B$

$\begin{pmatrix} 1+4 & 2+3 \\ 3+2 & 4+1 \end{pmatrix} = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$

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3. Compute $AB$

$\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 1\times2+2\times1 & 1\times0+2\times3 \\ 0\times2+1\times1 & 0\times0+1\times3 \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 1 & 3 \end{pmatrix}$

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4. Determine whether $A$ is invertible. If so, find $A^{-1}$

$|A| = 1\times4 - 2\times3 = 4-6 = -2 \neq 0$, so it is invertible.

$A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$

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5. Find the adjugate matrix of $A$

$A^* = \begin{pmatrix} 4 & -3 \\ -2 & 1 \end{pmatrix}$

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6. Write the elementary matrix that swaps the first and second rows of $A$

$E = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

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7. Find the rank of $A$

The two rows are proportional, so the rank is 1.

上一章节 Determinants Learn the concepts, properties, and calculation methods of determinants, laying the foundation for solving systems of linear equations. 下一章节 Systems of Linear Equations Understand the structure of solutions for homogeneous and nonhomogeneous systems, master Cramer's rule, fundamental solution sets, general solutions, and elementary row transformations.