Problem 32 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{ccc} 4 & -1 & 2 \\ 6 & -1 & 0 \\ 1 & -3 & 4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \(-26\).

Step 1: Identifying the Determinant Formula

For a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated using:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula helps us compute the determinant of the given matrix.

Step 2: Substitute Values into the Formula

Given the matrix:\[\begin{bmatrix} 4 & -1 & 2 \ 6 & -1 & 0 \ 1 & -3 & 4 \end{bmatrix}\]we have \( a = 4 \), \( b = -1 \), \( c = 2 \), \( d = 6 \), \( e = -1 \), \( f = 0 \), \( g = 1 \), \( h = -3 \), and \( i = 4 \). Substitute these into the determinant formula to get:\[4((-1)(4) - (0)(-3)) - (-1)((6)(4) - (0)(1)) + 2((6)(-3) - (-1)(1))\]

Step 3: Calculate Each Component

Now, calculate each part of the expression:- \( 4((-1)(4) - (0)(-3)) = 4(-4 - 0) = 4(-4) = -16\)- \( -(-1)((6)(4) - (0)(1)) = 1(24 - 0) = 24\)- \( 2((6)(-3) - (-1)(1)) = 2(-18 + 1) = 2(-17) = -34\)

Step 4: Sum the Calculated Values

Add the results from each component to find the determinant:\[\text{det}(A) = -16 + 24 - 34 = -26\]The determinant of the matrix is \(-26\).

Key Concepts

3x3 matrix

A 3x3 matrix is a square matrix consisting of three rows and three columns. These matrices are crucial in various mathematical computations. Each element in a 3x3 matrix typically represents a number located at the intersection of a specific row and column. Working with 3x3 matrices in mathematics helps in making more sophisticated calculations like solving systems of linear equations, performing transformations, or finding determinants. The structure of a 3x3 matrix can be represented as:

• First row: \(a, b, c\)
• Second row: \(d, e, f\)
• Third row: \(g, h, i\)

This type of matrix is frequently used in linear algebra due to its relatively compact yet comprehensive nature. Understanding how to manipulate and calculate these matrices is essential for further studies in mathematics and applied fields.

determinant formula

The determinant of a 3x3 matrix is a single value summarizing certain properties of the matrix. It can tell us if a matrix is invertible or singular, among other characteristics.The formula for finding the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This step-by-step calculation involves computing three 2x2 minors, then combining them with alternating signs. Here's how to break it down:

• Calculate the minor for each element of the first row (a, b, c).
• Use these minors to construct the determinant expression.
• Combine these calculations, remembering the alternating pattern of signs \(+, -, +\), as you integrate each minor calculation.

This formula is vital because it simplifies the otherwise complex task of determining important features of a matrix.

matrix algebra

Matrix algebra is a field of mathematics focused on operations involving matrices. It encompasses concepts such as addition, subtraction, multiplication, and finding determinants. Using matrix algebra, we can solve complex systems of linear equations, perform geometric transformations, and analyze data. Some key operations in matrix algebra include:

• Addition and Subtraction: Adding two matrices involves combining their individual elements, while subtraction is done element-wise as well.
• Multiplication: Multiplying matrices is more involved. The number of columns in the first matrix must align with the number of rows in the second matrix.
• Determinant Calculation: As shown earlier, finding the determinant helps in understanding certain properties, like invertibility.

Understanding these operations and concepts is crucial for more advanced mathematics fields like calculus and statistics. Matrix algebra provides tools that are foundational across disciplines, bridging gaps between theory and real-world applications.