In the previous article, we have seen how determinant decides whether a system of equation (read square matrix) has inverse, or it has a solution, only when the determinant is not zero. The determinant is obtained from the equation given below.

To know more about finding determinant in this way , read previous article. Here we will discuss about finding determinant by cross multiplication but before that let us understand the different notations used to represent determinants.

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### Notation For Determinants

There are several notation for determinants given by earlier mathematicians. Suppose represents a augmented matrix from a system of linear equations, then determinant of is given below.

```
Let the matrix be a 2 x 2 matrix.
Different ways to represent determinant of matrix
-- (1)
-- (2)
-- (3)
```

### Determinant as a Function

Imagine determinant to a function that take a square matrix as input and give a single value as output. For example, be a function where could be any real number. Similarly, is a function that matrix as input and give a determinant value . The determinant value is always integer because it is linear combination of integers, that is, all values are integers in the matrix.

**Determinant of Matrix**

If is a matrix with just one element, then its determinant is the same element.

**Example #1**

```
Let be a square matrix of order
Then the determinant of is
```

### Determinant of Matrix

The determinant of a matrix is obtained by performing cross multiplication. See the following figure.

**Example #2**

Let be a square matrix. Find the determinant of the matrix .

**Solution:**

```
Let the be 2 x 2 square matrix.
```

**Example #3**

Let be a square matrix of order . Find the determinant of the matrix .

**Solution:**

```
Let be a square matrix of order .
```

### Determinant Of Matrix

The determinant of a matrix is also possible through cross multiplication; Since we have a larger matrix we need to convert the larger matrix into smaller matrix to compute determinant. See figure below.

**Step 1: **Select the first row, first element and strike out rest of the elements from first row and first column. Then use to remaining element to create 2 x 2 matrix and find its determinant. See image below.

Each **selected element must be multiplied **with the respective 2 x 2 determinant obtained by **eliminating 1st row and the respective column of selected element**.

**Step 2: **Add all terms together and **assign a negative or positive sign** to each term.

```
If the selected element from top row (a b c) is . then row and column
Then the sign of first element
Similarly, second element from top row( a b c).
Third element from top row( a b c).
```

**Step 3: **Write down the determinant function obtained previously.

```
The determinant of matrix.
```

Let us see few examples of determinant of matrices.

**Example #4**

Find the determinant of the following matrix.

.

**Solution:**

```
Given matrix , the determinant is following function.
Therefore,
```

**Example #5**

Find the determinant of the following matrix below.

.

**Solution:**‘

```
Given matrix
The determinant is given by following function,
Therefore,
```

Note that the bigger the matrix , we must find more smaller matrices of size 2 x 2 and its determinant and then add all to get the determinant of original matrix. The process of finding the determinant of smaller matrices and making a matrix of determinants in called **matrix of minors. **

We shall discuss more about it in the next article.