Problem 56 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The determinant is 7.

Step 1: Identifying the Matrix

The given matrix is:\[\begin{bmatrix}1 & 0 & -3 \2 & -1 & 4 \-3 & 0 & 2\end{bmatrix}\]

Step 2: Determinant Formula for a 3x3 Matrix

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

Step 3: Substitute Values into the Formula

Substitute the values from the matrix into the formula:- \( a = 1, b = 0, c = -3 \)- \( d = 2, e = -1, f = 4 \)- \( g = -3, h = 0, i = 2 \)The formula becomes:\[1((-1)(2) - (4)(0)) - 0((2)(2) - (4)(-3)) + (-3)((2)(0) - (-1)(-3))\]

Step 4: Calculate Each Term

Calculate each smaller determinant:1. First term: \( 1((-1)(2) - (4)(0)) = 1(-2) = -2 \)2. Second term: \( 0((2)(2) - (4)(-3)) = 0 = 0 \)3. Third term: \( -3((2)(0) - (-1)(-3)) = -3(0 - 3) = -3(-3) = 9 \)

Step 5: Combine the Results

Add the results from each term to get the determinant:\[-2 + 0 + 9 = 7\]

Step 6: Conclusion

The determinant of the given matrix is 7.

Key Concepts

3x3 Matrix

A 3x3 matrix is a rectangular array consisting of three rows and three columns. It's often used in mathematics to solve systems of equations, perform transformations, and analyze linear relationships in 3-dimensional space. Each element in the matrix is placed in a specific position, identified by its row and column number. In our specific example, the matrix is as follows:

• The first row contains elements: 1, 0, -3.
• The second row has: 2, -1, 4.
• The third row consists of: -3, 0, 2.

Each position in the matrix can be crucial for determining properties like rank and determinant, which are important in various applications like physics and engineering.

Matrix Operations

Matrix operations are essential techniques in linear algebra, allowing us to manipulate and calculate important properties of matrices. Some key operations include addition, subtraction, multiplication, and finding the determinant or inverse of a matrix. Performing operations on matrices requires careful attention to their dimensions and conformity. In the context of a 3x3 matrix, determining the determinant is a common operation. The determinant offers information about a matrix such as whether it is invertible or not. Calculating the determinant involves a specific formula that multiplies components of the matrix, sums them up, and finds the resulting value. Understanding these operations helps you apply matrices to real-world problems and mathematical puzzles.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators that represent a particular value. In linear algebra, these expressions manifest when calculating determinants or when performing matrix operations.For a 3x3 matrix determinant, the expression involves multiple terms created by the elements of the matrix. Let's break one down:

• The formula for a determinant: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
• Inserting the values for the matrix elements, we develop a set of algebraic calculations.
• Each term in the formula represents a mini algebraic expression.

By calculating each term separately and then combining, we simplify the complex expression into a manageable form, such as the matrix determinant. This approach highlights the power and flexibility of algebra in solving linear algebra problems.