Problem 55 - Algebra 2

Question

Evaluate each determinant. $$ \left|\begin{array}{rrr}{-3} & {-3} & {1} \\ {-9} & {-2} & {3} \\ {5} & {-2} & {-1}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \(-14\).

Step 1: Identify the Determinant Formula for 3x3 Matrices

The determinant of a 3x3 matrix \( A \) with elements \( a_{ij} \) is calculated as follows:\[det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Step 2: Assign Matrix Elements to Variables

For the matrix \[ \begin{pmatrix} -3 & -3 & 1 \ -9 & -2 & 3 \ 5 & -2 & -1 \end{pmatrix} \]assign:- \( a_{11} = -3 \), \( a_{12} = -3 \), \( a_{13} = 1 \)- \( a_{21} = -9 \), \( a_{22} = -2 \), \( a_{23} = 3 \)- \( a_{31} = 5 \), \( a_{32} = -2 \), \( a_{33} = -1 \)

Step 3: Apply Determinant Formula Part 1

Evaluate the expression \( a_{11}(a_{22}a_{33} - a_{23}a_{32}) \): \[ a_{11}(a_{22}a_{33} - a_{23}a_{32}) = -3((-2)(-1) - (3)(-2)) = -3(2 + 6) = -3(8) = -24 \]

Step 4: Apply Determinant Formula Part 2

Evaluate the expression \( -a_{12}(a_{21}a_{33} - a_{23}a_{31}) \): \[ -a_{12}(a_{21}a_{33} - a_{23}a_{31}) = -(-3)((-9)(-1) - (3)(5)) = 3(9 - 15) = 3(-6) = -18 \]

Step 5: Apply Determinant Formula Part 3

Evaluate the expression \( a_{13}(a_{21}a_{32} - a_{22}a_{31}) \): \[ a_{13}(a_{21}a_{32} - a_{22}a_{31}) = 1((-9)(-2) - (-2)(5)) = 1(18 + 10) = 1(28) = 28 \]

Step 6: Compute the Determinant

Summing the results from steps 3, 4, and 5:\[det(A) = -24 - 18 + 28 = -14 \]

Step 7: Conclusion

The determinant of the given matrix is \(-14\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a grid of numbers arranged in three rows and three columns. Each number in the grid is called an element, which is typically denoted as \( a_{ij} \) where \( i \) is the row number and \( j \) is the column number. In our original exercise, the matrix given is:\[\begin{bmatrix}-3 & -3 & 1 \-9 & -2 & 3 \5 & -2 & -1 \end{bmatrix}\]This specific form allows us to perform various computations, like finding determinants or working with them in linear equations. These matrices are foundational in fields that utilize linear transformations, such as computer graphics and physics. Understanding how to navigate these matrices is a crucial skill in linear algebra courses and beyond. Whether you're dealing with rotations or scaling transformations, 3x3 matrices are an everyday tool in applied mathematics.

Matrix Operations

Matrix operations involve various techniques you can perform on matrices such as addition, subtraction, multiplication, and finding the determinant. Each operation has specific rules that must be followed to ensure accuracy.To find a determinant of a 3x3 matrix, for example, you follow a specific formula that involves taking the products of elements and their corresponding cofactors. The computation involves three main parts, each contributing to the total determinant:

• First, calculate \( a_{11} \) times the determinant of its minor matrix.
• Second, subtract \( a_{12} \) times the determinant of its minor matrix.
• Finally, add \( a_{13} \) times the determinant of its minor matrix.

This structured approach is essential for accurate outcomes when finding determinants. Practicing these operations helps cement your understanding and makes dealing with complex matrices more intuitive.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, and linear transformations, which often involve matrices. It enables the modeling and solving of systems of linear equations, which are equations where each term is either a constant or the product of a constant and a single variable. Matrices are used heavily in linear algebra because they offer a compact way to represent and manipulate linear transformations. Determinants, like the one we calculated in the original exercise, provide insight into the properties of a matrix, such as whether it is singular (non-invertible) or non-singular (invertible). Key concepts in linear algebra include:

• Vector spaces: The set of vectors where vector addition and scalar multiplication are defined.
• Eigenvalues and eigenvectors: Special values and corresponding vectors that reveal essential properties of a transformation.
• Rank and nullity: These properties offer information about the solutions of a matrix equation.

Linear algebra offers powerful tools crucial not just in mathematics, but in fields like engineering, computer science, and economics as well. Mastering these basics opens the door to exploring more advanced topics and real-world applications.