Problem 54 - Algebra 2

Question

Evaluate each determinant. $$ \left|\begin{array}{rrr}{2} & {8} & {-6} \\ {4} & {5} & {2} \\ {-3} & {-6} & {-1}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 52.

Step 1: Identify the Formula for a 3x3 Determinant

To evaluate a 3x3 determinant, we'll use the formula:\[\left|\begin{array}{ccc}a & b & c \d & e & f \g & h & i \end{array}\right| = a(ei - fh) - b(di - fg) + c(dh - eg)\]

Step 2: Substitute the Values into the Formula

Using the matrix provided:\[a = 2,\ b = 8,\ c = -6,\ d = 4,\ e = 5,\ f = 2,\ g = -3,\ h = -6,\ i = -1\]We substitute these into the formula for the determinant calculation.

Step 3: Calculate Each Minor Determinant

First, calculate each minor determinant:- For \( a \): \( ei - fh = (5 \times -1) - (2 \times -6) = -5 + 12 = 7 \)- For \( b \): \( di - fg = (4 \times -1) - (2 \times -3) = -4 + 6 = 2 \)- For \( c \): \( dh - eg = (4 \times -6) - (5 \times -3) = -24 + 15 = -9 \)

Step 4: Compute the Determinant

We substitute the calculated minor determinants back into the main formula:\[2(7) - 8(2) - 6(-9) = 14 - 16 + 54\]Simplifying gives:\[14 - 16 + 54 = 52\]

Step 5: Verify the Calculation

Double-check each arithmetic calculation:- \( 2 \times 7 = 14 \)- \( 8 \times 2 = 16 \)- \( -6 \times -9 = 54 \)Combine these results to ensure the total is correct:\[14 - 16 + 54 = 52\]

Key Concepts

Matrix Operations

Matrix operations are mathematical procedures that involve manipulating matrices or arrays of numbers. At the heart of these operations is the ability to perform additions, subtractions, multiplications, and finding determinants. Each operation follows specific rules due to the two-dimensional nature of matrices.

When working with matrices, some common operations include:

• Addition and Subtraction: Matrices can be added or subtracted if they are of the same dimensions. This involves adding or subtracting corresponding elements in each matrix.
• Scalar Multiplication: This operation involves multiplying each element of a matrix by a single number, known as a scalar.
• Matrix Multiplication: Unlike addition and subtraction, matrices can be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix. This process involves rows and columns and results in a new matrix.
• Finding Determinants: Especially for square matrices, determinants are crucial for understanding properties such as invertibility. The determinant is a scalar value that can be computed from a square matrix.

Applying these operations correctly allows us to solve various mathematical, scientific, and engineering problems efficiently.

3x3 Matrices

3x3 matrices are particularly important because they represent a manageable size that is also complex enough to demonstrate many key concepts in linear algebra. A 3x3 matrix has three rows and three columns, forming a square matrix.

3x3 matrices are often encountered in:

• Geometric Transformations: Used in rotations, scaling, and translations in three-dimensional space.
• Systems of Linear Equations: They can represent systems with three equations and three unknowns, creating a compact representation of the problem.
• Determinants: The calculation of 3x3 determinants is an essential skill. It involves applying a specific formula that incorporates the minor determinants of a matrix to arrive at a single scalar value.

Understanding the characteristics and operations of 3x3 matrices leads to deeper insights into more advanced matrix topics.

Minor Determinant Calculation

Minor determinant calculation is a crucial step when determining the determinant of a larger matrix, like a 3x3 matrix. The process involves computing the determinant of smaller 2x2 matrices, which are derived by eliminating one row and one column from the original matrix.

To calculate the minor determinant for a 3x3 matrix:

• Select a row or column to expand along.
• Identify the smaller 2x2 matrices, called "minors," by excluding the row and column of the element used to expand.
• Compute the determinant of these 2x2 minors using the formula: \[\begin{vmatrix}a & b \ c & d\end{vmatrix} = ad - bc\]
• Combine these calculations into the larger determinant formula which sums or subtracts these minor determinants appropriately.

This method reduces complex determinant calculations into manageable steps, ensuring accuracy and efficiency in solving matrix-related problems.