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.