Problem 17 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{5} & {1} & {0} \\ {0} & {2} & {-1} \\ {-2} & {-3} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

-3

Step 1: Write Down the Matrix

Write down the given matrix to prepare for calculating the determinant. The matrix is: \(\begin{bmatrix}5 & 1 & 0\ 0 & 2 & -1\ -2 & -3 & 1\end{bmatrix}\)

Step 2: Apply Determinant Formula for 3x3 Matrix

Use the formula for the determinant of a 3x3 matrix which is \(\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\). For our matrix, plug in the values which corresponds to: \(\text{det}(A) = 5(2*1 - (-1)*(-3)) - 1(0*1 - (-1)*(-2)) + 0(0*(-3) - 2*(-2))\).

Step 3: Calculate the Determinant

Calculate the determinant by performing the operations: \(\text{det}(A) = 5(2 - 3) - 1(0 - 2) + 0\), which simplifies to \(\text{det}(A) = 5(-1) - 1(-2)\), and after performing the multiplication, we get \(\text{det}(A) = -5 + 2\), resulting in \(\text{det}(A) = -3\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a square array of numbers consisting of three rows and three columns. It is a pivotal concept in fields like physics, engineering, and particularly in linear algebra. When we come across a 3x3 matrix, like the one in our exercise
\[\begin{bmatrix}5 & 1 & 0\0 & 2 & -1\-2 & -3 & 1\end{bmatrix}\],
we are looking at nine individual elements that can represent anything from coefficients in a system of equations to transformations in space.

Understanding the structure of such matrices is essential because it forms the foundation for more complex operations like matrix multiplication, inversion, and determining its determinant, which is a scalar value representing a certain property of the matrix. The determinant can tell us things like whether a matrix is invertible or what the volume scale factor of a geometric transformation is.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. One of the key operations in linear algebra is finding the determinant of a matrix, which gives valuable information about the matrix.

For instance, the determinant can determine if a set of linear equations has a unique solution—when it's non-zero—or if there are infinite solutions or no solution at all—when the determinant is zero. The determinant also helps to understand geometrical interpretations, like the area or volume scaled by a matrix when it acts on a geometrical shape. It's fascinating how such a compact representation of numbers can reveal so much about equations and transformations in space!

Matrix Operations

Matrix operations include a series of arithmetic procedures that can be conducted on matrices, including addition, subtraction, multiplication, and finding the inverse or determinant. When dealing with the determinant of a 3x3 matrix, its value is calculated using a particular formula that incorporates elements from each row and column.

Focusing on the Determinant

The term 'matrix operations' might suggest a wide range of processes, but let's narrow it down to the process of finding a determinant. It's crucial to execute the operations correctly to obtain the correct scalar. The determinant is often used to solve systems of linear equations (using Cramer's Rule), to find inverse matrices, and to understand properties like matrix singularity or invertibility. In our exercise, after applying the determinant formula, we conducted arithmetic operations systematically, emphasizing the importance of order and precision in matrix operations.