Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Explain why the determinant of the matrix is equal to zero. $$\left[\begin{array}{rrr} 2 & -4 & 5 \\ 1 & -2 & 3 \\ 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is equal to zero because all elements in its third row are zero. This property of determinants states that any matrix with a row or column of all zeros has a determinant of zero.

Step 1: Identify the matrix type

The matrix is a 3x3 square matrix, which means it has three rows and three columns.

Step 2: Examine the elements

It's noticeable that all the elements in the third row are zero. Therefore, based on the property of determinants, if any row or column of a matrix is made up of all zeroes, it will result in the determinant of the matrix being zero.

Step 3: Calculate the determinant

Though it's already clear that the determinant will be zero, as a formality, compute the determinant: \[ \begin{vmatrix} 2 & -4 & 5 \ 1 & -2 & 3 \ 0 & 0 & 0 \end{vmatrix} = 2 * \begin{vmatrix} -2 & 3 \ 0 & 0 \end{vmatrix} - (-4) * \begin{vmatrix} 1 & 3 \ 0 & 0 \end{vmatrix} + 5 * \begin{vmatrix} 1 & -2 \ 0 & 0 \end{vmatrix} = 0 \]

Key Concepts

Determinants Properties

Determinants serve as a fundamental tool in linear algebra and have properties that help us evaluate the characteristics of a matrix without extensive calculations. One of the essential properties of determinants states that if any row or column in a square matrix consists entirely of zeros, the determinant of that matrix will be zero. This property aligns with understanding that determinants reflect the 'volume' or 'scaling factor' encoded by a matrix, and a row or column of zeros implies there's no contribution in that particular dimension, collapsing the entire 'volume' to zero.

Other key properties include the fact that swapping two rows or columns changes the sign of the determinant, multiplying a row or column by a scalar multiplies the determinant by that scalar, and the determinant of a product of matrices equals the product of their determinants. Understanding these properties allows students to quickly assess certain matrix characteristics and simplifies determinant calculations significantly.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. It's used to represent many things in different fields, such as systems of linear equations in algebra, transformations in geometry, and tensors in physics. The elements within a 3x3 matrix can be numbers or more complex expressions depending on the context. Matrices of this size are common and important because they can describe 3D transformations and can be manipulated and used in matrix operations like addition, multiplication, and finding the inverse.

In terms of determinant calculation for a 3x3 matrix, there are specific patterns and methods, such as the rule of Sarrus or cofactor expansion, that can be applied to make finding the determinant more manageable.

Determinant Calculation

The calculation of a determinant for a 3x3 matrix can ideally be done using methods like the rule of Sarrus or the cofactor expansion, as mentioned earlier. The cofactor expansion, also known as Laplace's expansion, involves selecting a row or column, and for each element in that row or column, multiplying it by the determinant of the 2x2 submatrix that remains when the row and column containing that element are crossed out. Each of these products is then multiplied by \( (-1)^{i+j} \) where \( i \) and \( j \) correspond to the row and column of the element. Summing these up gives the total determinant.

For the given exercise with a zero row, the calculation is drastically simplified by the determinant property relating to zero rows, ensuring that our output determinant is, as proven, zero. In detail, when you apply the cofactor to the elements of the zero row in our example, each sub-determinant is multiplied by zero, leading to the whole determinant being zero, showcasing how determinant properties can simplify calculations.