Introduction of Operation with Matrices

1. Row Matrix: A matrix with only one row (1 × n). Example: $[1 \ 2 \ 3]$ (1 × 3).
2. Column Matrix: A matrix with only one column (m × 1). Example: $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$
3. Square Matrix: A matrix with the same number of rows and columns (n × n).
4. Identity Matrix: A square matrix with 1s on the main diagonal and 0s elsewhere, denoted by I. Example: $$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
5. Zero or Null Matrix: A matrix with all elements equal to 0. Example: $$0 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$
6. Diagonal Matrix: A square matrix with all non-diagonal elements equal to 0. Example: $$D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 8 \end{bmatrix}$$
7. Scalar Matrix: A diagonal matrix with all diagonal elements equal. Example: $$S = \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$$

Two matrices A and B are equal (A = B) if they have the same order and all corresponding entries are equal.

Addition and subtraction

Addition and subtraction are performed element-wise on matrices of the same order. The resulting matrix has the same order.

Example:

Properties of matrix addition

The sum of two m × n matrices is an m × n matrix.

Commutative: A + B = B + A

Associative: (A + B) + C = A + (B + C)

A + (-A) = 0 (zero matrix)

Properties of matrix subtraction

Not commutative: A - B ≠ B - A

The difference of two m × n matrices is an m × n matrix.

Not associative: (A - B) - C ≠ A - (B - C)

A - A = 0

Multiplication of a matrix by a scalar

Multiplying a matrix A by a scalar p results in a matrix where each element of A is multiplied by p.

Example:

Properties of scalar multiplication

r(A ± B) = rA ± rB

(r ± s)A = rA ± sA

(rs)A = r(sA) = s(rA)

Multiplication of a matrix by another matrix

Two matrices A (m × a) and B (b × n) can be multiplied if a = b. The resulting matrix AB has order m × n. Multiplication involves multiplying rows of A by columns of B.

Example:

Matrix multiplication is not commutative in general (AB ≠ BA).

Transpose of a matrix

The transpose of a matrix A (m × n), denoted by $A^T$, is an n × m matrix obtained by interchanging the rows and columns of A.

Example:

$(AB)^T = B^TA^T$