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# Determinant Formula

The determinant formula is used to find the determinant of a given matrix quickly. It is a mathematical object which is defined only for square matrices. A square matrix is a matrix in which the number of rows is equal to the number of columns. Let us learn about the determinant formula along with a few solved examples.

## What Is Determinant Formula?

The determinant of a matrix is a number defined only for square matrices. It is used in the analysis of linear equations and their solution. The determinant formula helps calculate the determinant of a matrix using the elements of the matrix.

### Determinant Formula

The determinant formula for 2 by 2 matrix that is \(\left [\begin{matrix}a & b\\c & d\end{matrix}\right]\) is given by:

D\(_{2\times2}\)_{ }= ad - bc

The determinant formula for 3 by 3 matrix is \(\left [\begin{matrix}a & b & c\\d & e & f\\ g & h & i\end{matrix}\right]\) is given by:

D\(_{3\times3}\)= a(ei-fh)-b(di-fg)+c(dh-eg)

## Applications of Determinant Formula

Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. Determinants also have wide applications in engineering, science, economics, and social science as well. Let’s now study the determinant of a matrix.

Let's take a quick look at a couple of examples to understand the determinant formula, better.

## Examples Using Determinant Formula

**Example 1: **Find the determinant of the 2x2 matrix below:

\(\left [\begin{matrix}2 & 3\\4 & 8\end{matrix}\right]\)

**Solution:**

To find: Determinant of the matrix.

Given:

a = 2; b = 3

c = 4; d = 8

Using determinant formula,

D\(_{2\times2}\) = ad - bc

Put the values,

D\(_{2\times2}\)= 2(8)-3(4)

=16-12

= 4

**Answer: Determinant of the matrix is 4.**

**Example 2:** Find the determinant of the 3x3 matrix below:

\(\left [\begin{matrix}6 & 1 & 1\\4 & -2 & 5\\ 2 & 8 & 7\end{matrix}\right]\) .

**Solution:**

To find: Determinant of the matrix

a = 6; b = 1; c = 1

d = 4; e = -2; f = 5

g = 2; h = 8; i = 7 (given)

Using determinant formula,

D\(_{3\times3}\) =a(ei - fh) - b(di - fg) + c(dh - eg)

=6(−2×7 − 5×8) − 1(4×7 − 5×2) + 1(4×8 − (−2×2))

= 6(−54) − 1(18) + 1(36)

= -306

**Answer: Determinant of the matrix is (-306).**

**Example 3: **Find the determinant of a 2x2 matrix,\(\left [\begin{matrix}4 & 1\\1 & 5\end{matrix}\right]\) .

**Solution:**

To find: Determinant of the matrix.

Given:

a = 4; b = 1

c = 1; d = 5

Using determinant formula,

D\(_{2\times2}\)= ad - bc

Put the values,

D\(_{2\times2}\)= 4(5)-1(1)

=20-1

=19

**Answer: The determinant of the matrix is 19.**

## FAQs on Determinant Formula

### What Is the Determinant Formula For a Given Matrix?

The determinant of a matrix is defined only for square matrices and helps to calculate the determinant of a matrix using the elements of the matrix.

- The determinant formula for 2 by 2 matrix, D\(_{2\times2}\) = ad - bc
- The determinant formula for 3 by 3 matrix, D\(_{3\times3}\) = a(ei-fh)-b(di-fg)+c(dh-eg)

### What Is the Special Feature Of the Determinant Formula?

The determinant of a matrix is defined only for square matrices and this property of the determinant formula makes it unique.

### How To Calculate the Determinant of a 2×2 Matrix Using Determinant Formula?

To calculate the determinant of a 2×2 matrix

- Step 1: Check if the given matrix is a square matrix that too a 2×2 matrix.
- Step 2: Identify all its rows and columns.
- Step 3: Put the values in the determinant formula, D\(_{2\times2}\)
_{ }= ad - bc

The determinant formula for 2 by 2 matrix that is D\(_{2\times2}\) is given by |ad - bc|

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