__Matrix__

__Matrix__

A **matrix** is simply a set of numbers arranged in a rectangular table.

There are several types of matrices, but the most commonly used are:

- Rows Matrix
- Columns Matrix
- Rectangular Matrix
- Square Matrix
- Diagonal Matrix
- Scalar Matrix
- Identity Matrix
- Triangular Matrix
- Null or Zero Matrix
- Transpose of a Matrix

**Row Matrix:**

A matrix is said to be a row matrix if it has only one row.

**Column Matrix:**

A matrix is said to be a column matrix if it has only one column.

**Rectangular Matrix:**

A matrix is said to be rectangular if the number of rows is not equal to the number of columns.

**Square Matrix:**

A matrix is said to be square if the number of rows is equal to the number of columns.

**Diagonal Matrix:**

A square matrix is said to be diagonal if at least one element of principal diagonal is non-zero and all the other elements are zero.

**Scalar Matrix:**

A diagonal matrix is said to be scalar if all of its diagonal elements are the same.

**Identity or Unit Matrix:**

A diagonal matrix is said to be identity if all of its diagonal elements are equal to one, denoted by I.

**Triangular Matrix:**

A square matrix is said to be triangular if all of its elements above the principal diagonal are zero **(lower triangular matrix)** or all of its elements below the principal diagonal are zero **(upper triangular matrix)**.

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**Null or Zero Matrix:**

A matrix is said to be a null or zero matrix if all of its elements are equal to zero. It is denoted by O.

**Transpose of a Matrix:**

Suppose A is a given matrix, then the matrix obtained by interchanging its rows into columns is called the transpose of A. It is denoted by A^{t}.

__Determinants of a matrix__

The determinant of a matrix is a **special number** that can be calculated from a square matrix.

For a 2×2 Matrix

__For a 2×2 matrix (2 rows and 2 columns): __

For a 3×3 Matrix

For a 3×3 matrix (3 rows and 3 columns):

__Adjoint of a Matrix__

Let A=[a_{ij}] be a square matrix of order n. The adjoint of matrix A is the transpose of the cofactor matrix of A. It is denoted by adj A. An adjoint matrix is also called an adjoint matrix.

__Example:__

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

__Inverse of a matrix__

The inverse of A is A^{-1} only when:

A × A^{-1} = A^{-1} × A = **I**

Sometimes there is no inverse at all.

__Formula:__

__Multiplication of a matrix__