Problem 20 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{rrr}6 & -1 & 2 \\ -4 & -3 & 5 \\ 1 & 9 & -1\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -317.

Step 1: Identify the Matrix

We have a 3x3 matrix for which we need to find the determinant. The matrix is: \[\begin{bmatrix}6 & -1 & 2 \-4 & -3 & 5 \1 & 9 & -1\end{bmatrix}\]

Step 2: Use the Rule of Sarrus

For a 3x3 matrix, the determinant can be calculated using the Rule of Sarrus, which involves multiplication of diagonals. For a matrix \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}\), the determinant is:\[\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Step 3: Substitute Matrix Values into the Formula

Substitute each element of the matrix into the determinant formula:Let \(a = 6\), \(b = -1\), \(c = 2\), \(d = -4\), \(e = -3\), \(f = 5\), \(g = 1\), \(h = 9\), \(i = -1\).The formula becomes:\[\text{det}(A) = 6((-3)(-1) - 5(9)) - (-1)((-4)(-1) - 5(1)) + 2((-4)(9) - (-3)(1))\]

Step 4: Calculate the Determinant

Now perform the arithmetic:1. Calculate each term: - For the first term: \((-3)(-1) - (5)(9) = 3 - 45 = -42\) - For the second term: \((-4)(-1) - (5)(1) = 4 - 5 = -1\) - For the third term: \((-4)(9) - (-3)(1) = -36 + 3 = -33\)2. Substitute back and compute: - \(6(-42) = -252\) - \(-(-1)(-1) = -1\) - \(2(-33) = -66\)3. Add them together:\( ext{det}(A) = -252 + 1 - 66 = -317\)

Key Concepts

Rule of Sarrus

To find the determinant of a 3x3 matrix efficiently, the Rule of Sarrus provides a straightforward approach. This rule is exclusive to 3x3 matrices and leverages the sum of products of diagonals. In the context of a 3x3 matrix like \[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}\]it involves:

• Finding the sum of the products of the diagonals from the upper left to lower right.
• Subtracting the sum of the products of the diagonals from the lower left to upper right.

To apply this, lay out the matrix and extend it to include the first two rows again beside the matrix. This helps visualize all diagonal paths. The determinant is then calculated by taking these diagonal paths, multiplying along them, and performing the additions and subtractions as outlined in the rule. This provides a reliable method for quickly deriving the determinant without extensive calculations.
Understanding this rule simplifies the process of finding determinants greatly.

matrix algebra

Matrix algebra is a branch of mathematics rich with operations similar to regular arithmetic but performed on matrices. Matrices are organized as arrays of numbers or symbols that can represent anything from coordinates to coefficients in equations. A fundamental operation within this algebra is the calculation of determinants. Determinants play a vital role as they indicate certain properties of matrices, such as invertibility.
In 3x3 matrices, determinants are critical as they define whether or not the matrix has an inverse. If the determinant is zero, the matrix is singular and non-invertible, meaning it cannot be "reversed" in operations. Conversely, a non-zero determinant indicates that the matrix is invertible.
Matrix algebra often involves various operations like addition, subtraction, multiplication, and finding inverses. Each operation follows specific rules:

• Addition/Subtraction: Performed element-wise between matrices of the same dimensions.
• Multiplication: Requires that the number of columns in the first matrix equals the number of rows in the second; the product, however, is not generally commutative.
• Inversion: Only possible if the matrix is square and has a non-zero determinant.

Understanding these operations builds a foundation for tackling more complex matrix-related problems.

arithmetic operations in determinants

Arithmetic operations within the context of determinants involve precise steps that assess the position of each element within the matrix. When calculating a determinant, as seen in the original exercise,
you must:

• Multiply specific elements based on their position in relation to others.
• Perform addition and subtraction of these products accurately, factoring in negative signs as dictated by their formulaic place.

The operations, while straightforward, require a careful eye for detail. The process generally includes:

• Identifying all terms that contribute to the determinant as per the Rule of Sarrus.
• Ensuring multiplication and any subsequent addition or subtraction respects proper arithmetic operations.
• Checking each calculation step for errors, especially paying attention to signs which can change the entire result.

Errors, especially in sign, can easily skew results, making practice and familiarity with the process vital. The goal is to become adept at recognizing patterns and performing these operations with ease, ensuring accuracy in mathematical endeavors.