Problem 18 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rrr}{7} & {3} & {-4} \\ {-2} & {9} & {6} \\ {0} & {0} & {0}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 0 because the matrix has a row of zeros.

Step 1: Understanding the Matrix

First, let's identify our 3x3 matrix: \[ \begin{bmatrix} 7 & 3 & -4 \ -2 & 9 & 6 \ 0 & 0 & 0 \end{bmatrix} \] Notice that the third row is entirely filled with zeros.

Step 2: Recognize the Zero Row Property

One important property of determinants is that if any row or column of a matrix is entirely composed of zeros, the determinant of that matrix is zero. In our matrix, the third row consists only of zeros.

Step 3: Apply the Zero Determinant Property

Based on the zero row property, we can directly state that the determinant of the matrix is zero. There's no need for further calculation using the cofactor expansion method or other determinant formulae.

Key Concepts

Zero Row Property

The zero row property is one of the simplifications you can use when working with determinants of matrices. This property states that if any row or column of a matrix is made up entirely of zeros, then the determinant of that matrix is automatically zero.
Why is this useful? Because it saves you time and effort! Instead of performing lengthy calculations, you can quickly identify matrices that have a zero row or column and conclude that their determinants are zero.
This property is particularly helpful in larger matrices where calculations can become more complex. So, always keep an eye out for those handy zero rows or columns.

3x3 Matrix

A 3x3 matrix is a square grid consisting of 3 rows and 3 columns of numbers. It can be represented in a neat, structured form often enclosed by brackets.
In the context of determinants, a matrix like the one in the exercise forms the basis for calculating the determinant, which is a specific scalar value. Determinants have multiple uses, including solving systems of linear equations, understanding eigenvalues, and more.
Here's a simple depiction of a 3x3 matrix:

• First row: numbers providing the coefficient of the first variable.
• Second row: numbers for the second variable.
• Third row: numbers for the third variable. If this row contains zeros, the determinant is instantly zero due to the zero row property.

Matrix Properties

Matrices have several fascinating properties that simplify complex mathematical operations. Knowing these can immensely aid your learning and problem-solving skills. Let's discuss some important matrix properties:

• Additive Property: You can add two matrices of the same size by adding their corresponding elements.
• Scalar Multiplication: Each element of a matrix can be multiplied by a constant scalar value, affecting the entire matrix uniformly.
• Transpose: Flipping a matrix over its diagonal, effectively turning its rows into columns and vice versa.
• Determinant Properties: These include properties like the zero row property we've discussed, as well as others like using cofactor expansion, and the product rule of determinants.

Understanding these properties not only aids in calculating determinants but also helps in many linear algebra applications. They make analyzing and manipulating matrices much more manageable.