Matrix Formula

Matrix

A matrix is a rectangular arrangement of numbers (real or Complex) which may be represented as

\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}

Matrix is enclosed by [] or ()

The compact form of the above matrix is represented by [a_ij] _m×n or A = [a_ij]

Order of a Matrix

The order of the matrix is the number of rows and columns present in the matrix.

If the matrix has m rows and n columns, then the order is m×n

Type of matrix

1. Row Matrix: A matrix with only one row is called a row matrix. It has 1 row and multiple columns.
2. Column Matrix: A matrix with only one column is called a column matrix. It has 1 column and multiple rows.
3. Square Matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix.
4. Diagonal Matrix: A square matrix where all the elements outside the main diagonal are zero is called a diagonal matrix.
5. Identity Matrix (I): A special diagonal matrix where all the diagonal elements are 1 and all other elements are 0 is called an identity matrix.
6. Zero Matrix (O): A matrix where all the elements are zero is called a zero matrix or a null matrix.
7. Scalar Matrix: A diagonal matrix where all the diagonal elements are the same is called a scalar matrix.
8. Symmetric Matrix: A square matrix that is equal to its transpose is called a symmetric matrix.
9. Skew-Symmetric Matrix: A square matrix whose transpose is equal to the negation of itself is called a skew-symmetric matrix.

Algebra of Matrix

1. Addition of Matrices
2. Subtraction of Matrices
3. Multiplication of a Matrix by a Scalar
4. Multiplication of Matrix

Let A = \begin{bmatrix}a & b \\c & d \end{bmatrix} and B = \begin{bmatrix}p & q \\r & s \end{bmatrix}

Addition of Matrix

A + B = \begin{bmatrix}a & b \\c & d \end{bmatrix} + \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}a + p & b + q \\c + r & d + s \end{bmatrix}

Subtraction of Matrix

A – B = \begin{bmatrix}a & b \\c & d \end{bmatrix} – \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}a - p & b - q \\c - r & d - s \end{bmatrix}

Multiplication of Matrix by a Scalar

kA = k \begin{bmatrix}a & b \\c & d \end{bmatrix} = \begin{bmatrix}ka & kb \\kc & kd \end{bmatrix}

Where k is the scalar value or constant number

Multiplication of Matrices

A×B=\begin{bmatrix}a & b \\c & d \end{bmatrix}×\begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}ap + br & aq+bs \\cp+dr & cq+ds \end{bmatrix}

B×A=\begin{bmatrix}p & q \\r & s \end{bmatrix}×\begin{bmatrix}a & b \\c & d \end{bmatrix} = \begin{bmatrix}pa+qc & pb+qd \\ra+sc & rb+sd \end{bmatrix}

Note: A×B ≠ B×A

Transpose of a Matrix

Let A be a matrix of order m × n. Then, the n × m matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A’ or A^T.

Properties of transpose

For any two matrices A and B of suitable orders.

1. (A’)’ = A
2. (A ± B)’ = A’ ± B’
3. (kA)’ = kA’
4. (AB)’ = B’A’
5. (ABC)’ = C’B’A

Symmetric and Skew-Symmetric Matrices

1. A square matrix A is said to be symmetric if A’ = A.
2. A square matrix A is said to be skew-symmetric if A’ = -A

Properties

If A is a square matrix, then

1. A + A’ is a symmetric matrix
2. A – A’ is a skew-symmetric matrix
3. The sum of symmetric and skew-symmetric matrix = ½ (A + A’) + ½ (A – A’)

If A and B are symmetric matrices of the same order, then

1. AB – BA is a skew-symmetric matrix and
2. AB + BA is symmetric matrix

Some special types of Matrix

Orthogonal Matrix: A square matrix of order n is said to be orthogonal if AA’ = I_n = A’A.

Properties of Orthogonal Matrix

1. If A is an orthogonal matrix, then A’ is also an orthogonal matrix.
2. For any two orthogonal matrices A and B. AB and BA is also orthogonal matrix.
3. If A is an orthogonal matrix, then A^-1 is also orthogonal matrix.