Question

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 3 & 2 & -4 \\ -1 & 5 & -3 \\ 0 & 1 & 0 \end{array}\right] $$

Short answer

The determinant of the matrix is 13. The method used for this exercise was the Sarrus Rule.

Step-by-step solution

Arrange the Matrix for Sarrus Rule

Sarrus Rule is a short-cut method for finding the determinant of a 3x3 matrix. The first step is to write the matrix and repeat the first two columns to its right.\n\n\[ \left[\begin{array}{rrrrr} \n3 & 2 & -4 & 3 & 2\ \n-1 & 5 & -3 & -1 & 5\ \n0 & 1 & 0 & 0 & 1 \n\end{array}\right] \]

Apply Sarrus Rule

Next, find the product of the diagonals going from top left to bottom right and sum them. This gives the positive sum. Then, find the product of the diagonals going from bottom left to top right and sum them. This gives the negative sum.\n\n Positive sum = (3*5*0) + (2*-3*0) + (-4*-1*1) = 0 + 0 + 4 = 4\n\n Negative sum = (0*5*-4) + (1*-3*3) + (0*-1*2) = 0 + (-9) + 0 = -9

Calculate the Determinant

Subtract negative sum from the positive sum.\n\n Determinant = Positive Sum - Negative Sum = 4 - (-9) = 13

Key concepts

Sarrus Rule

Understanding the Sarrus Rule can greatly simplify finding the determinant of any 3x3 matrix. It's a method that's exclusive to 3x3 matrices and isn't applicable to matrices of other sizes. Here's a concise explanation of how it works.

Imagine you have a 3x3 matrix. To apply the Sarrus Rule, you write down the matrix and then, as a visual aid, you repeat the first two columns on the right-hand side of the matrix. The purpose of this repetition is not to change the matrix itself but to help us visualize the diagonals that we will use in the calculation.

Next, you take the sum of the products of the diagonals that go from the top left to the bottom right. These will form your positive sum. Conversely, you sum the products of the diagonals that go from the bottom left to the top right, which gives you the negative sum. The determinant of the matrix is the difference between these two sums. This approach allows for a quick mental calculation without having to memorize any complex formulas.

3x3 Matrix

A 3x3 matrix consists of nine elements arranged in three rows and three columns. It is a very common matrix size in linear algebra, often used to solve systems of equations, perform transformations in space, and more. The elements of a matrix can represent coefficients in linear equations, pixel values in digital images, or any number of other quantitative attributes depending on the context.

When working with such matrices, various operations can be performed, including addition, multiplication, and finding inverses. However, the Sarrus Rule specifically deals with finding the determinant, which is a scalar value that can give us information about the matrix, such as whether it's invertible or the volume distortion factor of the linear transformation the matrix represents.

Matrix Operations

Matrix operations are foundational to the study of linear algebra and have a wide array of applications in mathematics, science, engineering, statistics, and other fields.

Some of the basic matrix operations include:

• Addition: Summing corresponding elements from two matrices of the same size.
• Scalar Multiplication: Multiplying every element in a matrix by a scalar value.
• Matrix Multiplication: Combining two matrices to form a new matrix whose elements are calculated through a specific summation process.
• Transpose: Flipping a matrix over its diagonal, effectively switching rows with columns.
• Inverse: Finding a matrix that, when multiplied by the original, yields the identity matrix, assuming the original matrix is invertible.

In the context of determinants and Sarrus Rule, we're focused on a specific scalar operation that condenses all the information of a 3x3 matrix into a single number. This number greatly influences other matrix operations, such as calculating inverses, as only matrices with non-zero determinants have inverses.