Determinant of 3 x 3 Matrix calculator

About MathCrave Determinant of 3 x 3 Matrix

The determinant is a special number that can be calculated for square matrices, and it provides important information about the properties of the matrix. The determinant of a 3x3 matrix can be calculated using various methods such as cofactor expansion, row operations, or using the properties of determinants.

CoFactor

Cofactor expansion: Also known as expansion by minors, it is a method used to calculate the determinant of a square matrix by recursively expanding it along a row or a column using the cofactor of each element.

Row operations

These are operations that can be performed on the rows of a matrix to manipulate it without changing its determinant. The three types of row operations are:
1. Swapping two rows.
2. Multiplying a row by a non-zero constant.
3. Adding a multiple of one row to another row.

Using the Properties of Determinants

This refers to applying the various properties and rules of determinants to simplify the calculation. Some of the properties include:

1. Multiplying a row (or column) of a matrix by a constant multiplies the determinant by the same constant.
2. If two rows (or columns) of a matrix are interchanged, the determinant changes sign.
3. Adding a multiple of one row (or column) to another row (or column) does not change the determinant.
4. If a matrix has a row (or column) consisting of all zeros, then its determinant is zero.

Understanding the Formula for Determinant of 3 x 3 Matrix

The determinant of a 3x3 matrix can be found using the following formula:

• det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Here, A represents the 3x3 matrix, and a, b, c, d, e, f, g, h, and i are the elements of the matrix written in the following form:

• A =

• |a b c|
  |d e f|
  |g h i|

Detailed Guide to Solve Determinant of 3 x 3 Matrix

To solve the determinant of a 3x3 matrix, follow these steps:

Step 1: Write down the matrix. Let's say our matrix is A, and it is given by:

• A =

• | a b c |

• | d e f |

• | g h i |

Step 2: Start with the first element in the first row (a). Multiply this element by the determinant of the 2x2 matrix that is obtained by removing the row and column that contain it.

Step 3: The determinant of a 2x2 matrix can be found using the following formula:

• det(B) = b i - c h

Step 4: Multiply the result in step 3 by (-1)^n, where n is the position of the element in the first row (starting from 1). In our case, since a is in the first position,

• (-1)^1 = 1.

Step 5: Add this result to the next element in the first row (b). Multiply it by the determinant of the 2x2 matrix obtained by removing the row and column that contain it.

Step 6: Repeat steps 3, 4, and 5 for the remaining elements in the first row (c).

Step 7: Add up the results from steps 4 and 6 to find the determinant of the 3x3 matrix.

Determinant of 3 x 3 Matrix Worked Example

Example:

A =

| 2 3 1 |

| 0 -1 4 |

| 5 -2 3 |

Step 1: Write down the matrix A.

Step 2: Start with the first element in the first row (2). Multiply it by the determinant of the 2x2 matrix:

• | -1 4 |

• | -2 3 |

Step 3: Find the determinant of the 2x2 matrix:

• (-1 3) - (4 -2) = -3 + 8 = 5.

Step 4: Multiply

5 by (-1)^1 = 5 * 1 = 5.

Step 5: Add this result to the next element in the first row (3). Multiply it by the determinant of the 2x2 matrix:

• | 0 4 |

• | 5 3 |

Step 6: Find the determinant of the 2x2 matrix:

• (0 3) - (4 5) = 0 - 20 = -20.

Step 7: Add 5 and -20 together:

• 5 + (-20) = -15.

Step 8: Finally, add this result to the next element in the first row (1). Multiply it by the determinant of the 2x2 matrix:

• | 0 -1 |

• | 5 -2 |

Step 9: Find the determinant of the 2x2 matrix:

• (0 -2) - (-1 5) = 0 + 5 = 5.

Step 10: Add -15 and 5 together:

-15 + 5 = -10.

So, the determinant of the matrix A is

-10.

Determinant of 3 x 3 Matrix Worked Example 2

Let's find the determinant of the 3x3 matrix A:

A =

|2 1 3|
|0 4 1|
|-1 2 0|

Step 1: Write down the matrix A:

A = |2 1 3|
|0 4 1|
|-1 2 0|

Step 2: Compute the determinants of the 2x2 matrices:

det(A1) = |4 1| = (4*0) - (1*2) = -2
|2 0|

det(A2) = |2 3| = (2*0) - (3*-1) = 3
|-1 0|

det(A3) = |2 1| = (2*2) - (1*-1) = 5
|0 4|

Step 3: Substituting the values into the formula:

det(A) = 2(-2) - 1(3) + 3(5) = -4 - 3 + 15 = 8

Thus, the determinant of matrix A is 8.

Theorem Binding Determinant of 3 x 3 Matrix

One important mathematical theorem related to matrix determinants is Cramer's Rule. According to Cramer's Rule, given a system of linear equations represented by the matrix equation Ax = b, where A is a square matrix of coefficients, x is a column matrix of variables, and b is a column matrix of solutions, the determinant of matrix A must be non-zero for the system to have a unique solution.

Practice Questions on Determinant of 3 x 3 Matrix

• 1. Find the determinant of the matrix:
  [2 1 3]
  [4 2 1]
  [5 0 2]
  
  2. Calculate the determinant of the matrix:
  [3 1 5]
  [2 4 7]
  [9 0 2]
  
  3. Determine the determinant of the matrix:
  [1 2 3]
  [0 1 4]
  [2 3 1]
  
  4. Find the determinant of the matrix:
  [2 1 3]
  [1 4 2]
  [3 2 1]
  
  5. Calculate the determinant of the matrix:
  [-2 5 3]
  [1 4 1]
  [0 3 2]
  
  6. Determine the determinant of the matrix:
  [1 2 1]
  [3 0 2]
  [0 1 4]
  
  7. Find the determinant of the matrix:
  [1 2 3]
  [4 5 6]
  [7 8 9]
  
  8. Calculate the determinant of the matrix:
  [2 4 5]
  [1 3 2]
  [0 2 1]
  
  9. Determine the determinant of the matrix:
  [1 2 3]
  [-2 1 4]
  [0 3 1]
  
  10. Find the determinant of the matrix:
  [4 0 1]
  [2 3 5]
  [7 2 0]