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# Determinant Calculator

Determinant Calculator is an online tool that helps to find the determinant of a 3 x 3 square matrix using the determinant formula. We can find the determinant of square matrices only.

## What is Determinant Calculator?

Determinant Calculator helps to compute the determinant of a 3 x 3 square matrix. When we find the sum of the products of the elements of a square matrix according to some prescribed rule, the quantity so obtained is called a determinant. To use the * determinant calculator*, enter the values in the input boxes.

### Determinant Calculator

NOTE: Enter the values up to 3 digits only.

## How to Use the Determinant Calculator?

Please follow the steps below to find the determinant of the 3 x 3 matrix using the online determinant calculator.

**Step 1:**Go to Cuemath's online determinant calculator.**Step 2:**Enter the elements of the matrix in the given input boxes.**Step 3:**Click on the**"Calculate"**button to find the determinant.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Determinant Calculator Work?

A determinant can be thought of as a function that takes the elements of a square matrix as inputs and outputs a single value. Determinants are scalar quantities. The determinant of a square matrix is used to find the inverse of that matrix. Furthermore, we require determinants if we solve linear equations using the matrix inversion method. The steps to find the determinants of matrices are given below:

1. 2 x 2 Matrix

Let \(A_{2\times 2}\) = \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\)

The formula to calculate the determinant is as follows:

\(\begin{vmatrix}A_{2\times 2} \end{vmatrix}\) = (a x d) - (b x c)

2. 3 x 3 Matrix

Let \(A_{3\times 3}\) = \(\begin{bmatrix} a & b & c\\ d& e& f\\ g& h & i \end{bmatrix}\)

**Step 1:** We multiply the anchor numbers with the respective square sub matrix.

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = a . \(\begin{vmatrix} e & f\\ h & i \end{vmatrix}\) - b . \(\begin{vmatrix} d & f\\ g & i \end{vmatrix}\) + c . \(\begin{vmatrix} d & e\\ g & h \end{vmatrix}\)

**Step 2:** Using the determinant formula of the 2 x 2 matrix we solve the expression in step 1 as given below.

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = a(ei - fh) - b(di - fg) + c(dh - eg).

## Solved Examples on Determinants

**Example 1:** Find the determinant of following matrix \(\begin{bmatrix} 3 & 4 & 5\\ 1& 2& 3\\ 2& 0 & 9 \end{bmatrix}\) and verify it using the online determinant calculator.

**Solution:**

Given : \(A_{3\times 3}\) = \(\begin{bmatrix} 3 & 4 & 5\\ 1& 2& 3\\ 2& 0 & 9 \end{bmatrix}\)

According to the formula,

\(A_{3\times 3}\) = \(\begin{bmatrix} a & b & c\\ d& e& f\\ g& h & i \end{bmatrix}\)

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = a(ei - fh) - b(di - fg) + c(dh - eg)

Substititing these values we get

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = 3 (2 x 9 - 0 x 3) - 4(1 x 9 - 0 x 3) + 5(1 x 0 - 2 x 2)

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = 22

**Example 2:** Find the determinant of following matrix \(\begin{bmatrix} -1 & 2 & -4\\ 7.2& 6& 2.3\\ -5& 1.3 & 3 \end{bmatrix}\) and verify it using the online determinant calculator.

**Solution:**

Given : \(A_{3\times 3}\) = \(\begin{bmatrix} -1 & 2 & -4\\ 7.2& 6& 2.3\\ -5& 1.3 & 3 \end{bmatrix}\)

According to the formula,

\(A_{3\times 3}\) = \(\begin{bmatrix} a & b & c\\ d& e& f\\ g& h & i \end{bmatrix}\)

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = a(ei - fh) - b(di - fg) + c(dh - eg)

Substititing these values we get

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = -1 (6 x 3 - 2.3 x 1.3) - 2(7.2 x 3 - 2.3 x (-5)) - 4(7.2 x 1.3 - 6 x (-5))

\(\begin{vmatrix}A_{3\times 3} \end{vmatrix}\) = -238.65

Similarly, you can try the determinant calculator to find the determinants for the following:

- \(\begin{bmatrix} 3.2 & -2 & 3\\ 1& 0& 4\\ -1& -7 & 4.7 \end{bmatrix}\)
- \(\begin{bmatrix} 8.2 & -3 & 5\\ 1.9& 2.8& -3\\ 9& 1 & 1 \end{bmatrix}\)

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