How to Find Determinant of 3×3 Matrix

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Determinants play a crucial role in linear algebra, and they are used in various applications, such as solving systems of linear equations, calculating the area and volume of geometric objects, and more. A determinant is a scalar value that is associated with a square matrix. It is used to determine whether the matrix is invertible or not, and to calculate its inverse.

In this article, we will discuss how to find the determinant of a 3×3 matrix. We will start by defining what a determinant is, and then we will introduce the formula for calculating the determinant of a 3×3 matrix. We will then discuss the different methods for finding the determinant of a 3×3 matrix, including using cofactor expansion, using row operations, and using the diagonal rule. Finally, we will provide some examples and answer some frequently asked questions about finding the determinant of a 3×3 matrix.

What is a determinant?

A determinant is a scalar value that is associated with a square matrix. It is denoted by det(A) or |A|, where A is the matrix. The determinant of a matrix is a real number that can be calculated using various methods. The determinant is a useful tool for solving systems of linear equations, calculating the inverse of a matrix, and more.

Formula for calculating the determinant of a 3×3 matrix:

The formula for calculating the determinant of a 3×3 matrix is:

|A| = a11(a22a33 – a23a32) – a12(a21a33 – a23a31) + a13(a21a32 – a22a31)

where A is the 3×3 matrix:

| a11 a12 a13 | | a21 a22 a23 | | a31 a32 a33 |

Methods for finding the determinant of a 3×3 matrix:

There are several methods for finding the determinant of a 3×3 matrix. We will discuss three methods in this article: using cofactor expansion, using row operations, and using the diagonal rule.

Method 1: Using cofactor expansion

The cofactor expansion method involves expanding the determinant along a row or a column of the matrix. The cofactor of an element in a matrix is the determinant of the submatrix obtained by deleting the row and column that contain the element.

To find the determinant of a 3×3 matrix using cofactor expansion, we can choose any row or column and expand along it. Let’s assume we choose the first row. Then the formula becomes:

|A| = a11|A11| – a12|A12| + a13|A13|

where A11, A12, and A13 are the 2×2 submatrices obtained by deleting the first row and the column containing the corresponding element. The signs alternate in the formula, as shown above.

To find the determinant of the submatrices, we can use the formula we just discussed. For example, to find the determinant of A11, we can use the formula:

|A11| = a22a33 – a23a32

Similarly, we can find the determinants of A12 and A13. Once we have these determinants, we can substitute them into the formula and calculate the determinant of the 3×3 matrix.

Method 2: Using row operations

The row operation method involves performing elementary row operations on the matrix to simplify it and then using the diagonal rule to find the determinant.

To find the determinant of a 3×3 matrix using row operations, we can follow these steps:

  1. Swap two rows to make the leading coefficient in the first column nonzero.
  2. Multiply the first row by a scalar to make the leading coefficient
  1. Subtract a multiple of the first row from the second and third rows to make the entries below the leading coefficient in the first column zero.
  2. Repeat steps 1-3 with the second column and the third column until the matrix is in upper triangular form.
  3. Use the diagonal rule to calculate the determinant.

The diagonal rule states that the determinant of an upper triangular matrix is the product of the diagonal entries. Therefore, once the matrix is in upper triangular form, we can simply multiply the diagonal entries to find the determinant.

Method 3: Using the diagonal rule

The diagonal rule method involves using the pattern of signs in the formula for the determinant to simplify the calculation. To use this method, we simply multiply the diagonal entries of the matrix, and then we add the products of the entries that are not on the main diagonal, with appropriate signs.

To be more precise, the formula for the determinant of a 3×3 matrix using the diagonal rule is:

|A| = a11a22a33 + a12a23a31 + a13a21a32 – a31a22a13 – a32a23a11 – a33a21a12

In this formula, the terms with positive signs correspond to the products of the entries on the main diagonal, and the terms with negative signs correspond to the products of the entries off the main diagonal.

Examples:

Let’s consider some examples to illustrate the methods for finding the determinant of a 3×3 matrix.

Example 1: Find the determinant of the matrix A =

| 1 2 3 | | 4 5 6 | | 7 8 9 |

Using cofactor expansion:

|A| = 1|A11| – 2|A12| + 3|A13|

where A11 =

| 5 6 | | 8 9 |

A12 =

| 4 6 | | 7 9 |

A13 =

| 4 5 | | 7 8 |

We can calculate the determinants of these submatrices as follows:

|A11| = 59 – 68 = -3 |A12| = 49 – 67 = 6 |A13| = 48 – 57 = -3

Substituting these values into the formula, we get:

|A| = 1*(-3) – 26 + 3(-3) = -6

Therefore, the determinant of the matrix A is -6.

Using row operations:

We can perform row operations on the matrix to simplify it and bring it into upper triangular form. We can start by subtracting 4 times the first row from the second row and 7 times the first row from the third row, to get:

| 1 2 3 | | 0 -3 -6 | | 0 -6 -12 |

Next, we can divide the second row by -3 and the third row by -6, to get:

| 1 2 3 | | 0 1 2 | | 0 1 2 |

Finally, we can subtract the second row from the third row, to get:

| 1 2 3 | | 0 1 2 | | 0 0 0 |

The matrix is now in upper triangular form, and we can use the diagonal rule to calculate the determinant. The diagonal entries are 1, 1, and 0, so the determinant is:

|A| = 110 + 220 + 310 – 321 – 201 – 013 = -6

Therefore, we get the same result as before.

Using the diagonal rule:

We can directly apply the formula for the determinant using the diagonal rule:

|A| = 159 + 267 + 348 – 357 – 261 – 943 = -6

Again, we get the same result as before.

Example 2: Find the determinant of the matrix B =

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

Using cofactor expansion:

|B| = 2|B11| – 3|B12| + 4|B13|

where B11 =

| 0 -1 | | 1 2 |

B12 =

| 1 -1 | | 3 2 |

B13 =

| 1 0 | | 3 1 |

We can calculate the determinants of these submatrices as follows:

|B11| = 02 – (-1)1 = 1 |B12| = 12 – (-1)3 = 5 |B13| = 11 – 03 = 1

Substituting these values into the formula, we get:

|B| = 21 – 35 + 4*1 = -11

Therefore, the determinant of the matrix B is -11.

Using row operations:

We can start by subtracting 2 times the first row from the second row and 3 times the first row from the third row, to get:

| 2 3 4 | | 0 -6 -9 | | 0 -8 -10 |

Next, we can add 4 times the second row to the third row, to get:

| 2 3 4 | | 0 -6 -9 | | 0 0 -46 |

The matrix is now in upper triangular form, and we can use the diagonal rule to calculate the determinant. The diagonal entries are 2, -6, and -46, so the determinant is:

|B| = 2*(-6)*(-46) = 552

Using the diagonal rule:

We can directly apply the formula for the determinant using the diagonal rule:

|B| = 202 + 3*(-1)3 + 411 – 401 – 2(-1)3 – 01*2 = -11

Again, we get the same result as before.

Frequently asked questions:

What is the determinant of a matrix?

The determinant of a matrix is a scalar value that is associated with a square matrix. It is used to determine whether the matrix is invertible or not, and to calculate its inverse.

How do you find the determinant of a 3×3 matrix?

There are several methods for finding the determinant of a 3×3 matrix, including using cofactor expansion, using row operations, and using the diagonal rule.

What is the formula for the determinant of a 3×3 matrix?

The formula for the determinant of a 3×3 matrix is:

|A| = a11(a22a 33 – a23a32) – a12(a21a33 – a23a31) + a13(a21a32 – a22a31)

where A is the 3×3 matrix:

| a11 a12 a13 | | a21 a22 a23 | | a31 a32 a33 |

Can you use any row or column to find the determinant of a 3×3 matrix using cofactor expansion?

Yes, you can choose any row or column to expand the determinant using cofactor expansion. The signs alternate in the formula, depending on whether the indices of the row and column are even or odd.

How do you use the diagonal rule to find the determinant of a 3×3 matrix?

To use the diagonal rule, you multiply the diagonal entries of the matrix and then add the products of the entries that are not on the main diagonal, with appropriate signs.

What is the relationship between the determinant of a matrix and its inverse?

A matrix is invertible if and only if its determinant is nonzero. The inverse of a matrix can be calculated using the formula:

A^-1 = (1/|A|) * adj(A)

where adj(A) is the adjugate of A, which is obtained by taking the transpose of the matrix of cofactors.

Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative, positive, or zero. The sign of the determinant depends on the pattern of signs in the formula for the determinant.

What is the geometric interpretation of the determinant of a 3×3 matrix?

The absolute value of the determinant of a 3×3 matrix is equal to the volume of the parallelepiped formed by the vectors corresponding to the rows or columns of the matrix.

How can you check your answer when finding the determinant of a 3×3 matrix?

You can check your answer by calculating the determinant using different methods and comparing the results. You can also use software or online calculators to verify your answer.

What are some applications of determinants in mathematics and science?

Determinants are used in various applications, such as solving systems of linear equations, calculating the area and volume of geometric objects, finding eigenvalues and eigenvectors of matrices, and more. They are also used in physics, engineering, economics, and other fields.

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