Matrices form an important tool in the study of finite-dimensional vector space. Determinants form an important tool in the study of matrices.

**Introduction to Operations with Matrices:**

Matrices are rectangular array items which are represented in rows and columns. Data in the matrices are called entry or an element. The matrix looks like

`[[7,2,1,3],[8,3,9,1]]`

The row of matrix is the horizontal line and column of matrix is the vertical line.

Operations on matrices are :

- Addition
- Subtraction
- Multiplication
- Transpose

## Addition and subtraction of matrices

**Addition of Matrices:**

- The sum of two matrices should be of same order
- Sum of two matrices is obtained by adding the corresponding entries.

**Subtraction of Matrices:**

- The sum of two matrices should be of same order
- Sum of two matrices is obtained by subtracting the corresponding entries.

**Ex:** ** Let us consider A = `[[3,2,1],[2,1,4],[1,5,3]]`**`[[2,3,1],[5,2,6],[3,7,1]]`

**Sol: **Addition of matrices is done as shown below

** Step I: ** A + B = `[[3+2,2+3,1+1],[2+5,1+2,4+6],[1+3,5+7,3+1]]`

**Step II: ** = `[[5,5,2],[7,3,10],[4,12,4]]`

Subtraction of matrices is done as shown below,

**Step I: ** A – B = `[[3-2,2-3,1-1],[2-5,1-2,4-6],[1-3,5-7,3-1]]`

** Step II: ** = `[[1,-1,0],[-3,-1,-2],[-2,-2,2]]`

## Multiplication and transpose of Matrices:

**Multiplication of matrics:**

There are two types multiplication method done in matrices. They are

**Scalar multiplication:**Multiplication of a matrix by a scalar `alpha` is done by multilpying each entry by `alpha` .

** Ex:** **Let us consider the matrix A = `[[4,2,1],[3,5,2],[4,6,3]]` and scalar `alpha = 2` **

**Sol:** Scalar multiplication is calculated by

** Step I:** `alpha` A = `[[2*4,2*2,2*1],[2*3,2*5,2*2],[2*4,2*6,2*3]]`

**Step II:** 2A = `[[8,4,2],[6,10,4],[8,12,6]]`

**Matrix multiplication**: Unless the number of column of first matrix and row of second matrix are similar, we cannot perform the matrix multiplication.

Ex: ** Let us consider the matrix A = `[[2,3,4],[1,2,3],[4,1,2]]` and B = `[[1,2,3],[2,2,1],[1,2,3]]` .**

Sol: Matrix multiplication is calculated by :

** Step I:** A * B = `[[2+6+4,4+6+8,6+3+12],[1+4+3,2+4+6,3+2+9],[4+2+2,8+2+4,12+1+6]]`

** Step II:** = `[[12,18,21],[8,12,14],[6,14,19]]`

**Transpose: ** In Transpose, the row and column of the matrix is interchanged to column and row respectively.

** Ex: ** ** A = `[[1,2,3,4],[5,6,7,8]]` **

** Sol: ** Transpose of the matrix A is

**Step I: ** A^{T }= `[[1,5],[2,6],[3,7],[4,8]]`