The determinant of a matrix could be a special number that may be calculated from a square matrix. Determinants are like matrices, however, done up in absolute-value bars rather than square brackets. The determinant of a matrix could be a scalar property of the matrix. Only sq. matrices have determinants. If there is a matrix A then its determinant is written by taking numbers of elements and putting them within absolute-value bars rather than sq. brackets. The determinant is viewed as a result whose input could be a matrix and whose output could be a single number.

## Uses of Determinant

- Determinants are useful as a result of they tell us whether a matrix is inverted or not.
- It plays a vital role in solving the linear equation.
- The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
- Determinant has wide application in engineering, science, economics, social science, etc.

To every square matrix of order , we can associate a number [real or complex] called determinant of the square matrix A, where element of A. It is denoted by |A| or det A.

If then the determinant of A is written as = det (A).

**Note **

- For matrix A, |A| is read as the determinant of A and not modulus of A.
- Only square matrices have determinants.

## Determinant of a 2*2 matrix

Suppose A is a matrix of order 2*2 matrix such as : the the determinant of matrix A would be |A| = =

**Example 1: **Find the determinant of matrix

**Solution: ** = 4*5 – 3*2 = 20-6 = 14

**Example 2: **Find the determinant of matrix

**Solution: ** = 1*(-3) – 5*2 = -3 – 10 = -13

## Determinant of a 3*3 matrix

Suppose A is a matrix of order 3*3 matrix such as then the determinant of matrix A would be = – +

The element a1,b1,c1 of first row are in the expression with alternatively positive and negative sign and each element is multiplied by a each element is multiplied by a certain determinant of order 2.

**Example 3: **Let . Find determinant.

**Solution : ** – + = 4 (-4-1) + 3 (-2-3) + 2 (1-6) = -20-15-10 = -45

**Example 4: **Let

**Solution: ** – + = – + =

## Properties of Determinants

- The value of determinant remains unchanged if its rows and columns are interchanged.
- If any 2 rows or columns are being interchanged the there be the change in sign of determinants.
- If any 2 rows or columns are being identical then the value of determinant would be zero.
- If the element of a row or column is being multiplied by a scalar then the value of determinant also become a multiple of that constant.
- Two determinant can be added if they have 2 identical rows or columns.

## Exercise

- Find determinant of matrix
- Find determinant of matrix
- Find the value of x if = 0
- Evaluate
- Prove that = a(b-c) (a-b)

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