Question

Evaluate the determinant(s) to verify the equation. \(\left|\begin{array}{ll}w & c x \\ y & c z\end{array}\right|=c\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|\)

Short answer

The equation is verified, since the determinant of both matrices are equal when multiplied by the scalar c.

Step-by-step solution

Calculate the determinant of the left matrix

To calculate the determinant of a 2x2 matrix, do the following: Multiply the top left and bottom right elements, then subtract the product of the top right and bottom left elements. For the left matrix, this would be: \[ (w)(cz) - (cx)(y) = c(wz - xy) \]

Calculate the determinant of the right matrix

Applying the same process to the matrix on the right side of the equation gives: \[ c[(w)(z) - (x)(y)] = c(wz - xy) \]

Compare the determinants of both matrices

Now, you just need to compare the two results: If \( c(wz - xy) = c(wz - xy) \), the initial equation is verified.

Key concepts

Matrix Algebra

Matrix algebra is an essential branch of mathematics that deals with the study of matrices and the various operations that can be performed on them. It is a vital tool in linear algebra and has applications across different fields such as physics, economics, engineering, and computer science.

Considering our exercise, when working with matrices, it is important to understand basic operations such as addition, subtraction, multiplication, and finding the determinant. The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether it is invertible or if its vectors are linearly independent.

When evaluating the determinant of a 2x2 matrix, as shown in our step by step solution, we use a specific formula that helps us find this scalar. This process is a fundamental part of matrix algebra which leads to understanding more complex matrix operations. Matrix algebra forms the groundwork for advancing in linear algebra, where you begin to solve systems of equations, find eigenvalues, and work with vector spaces.

Properties of Determinants

Understanding the properties of determinants is crucial for simplifying calculations and grasping deeper concepts in linear algebra. Determinants have several important properties that can make dealing with matrices more manageable.

Firstly, the determinant of a matrix remains unchanged if a multiple of one row is added to another. Secondly, if two rows or columns of a matrix are identical or proportional, its determinant is zero, indicating that the matrix does not have an inverse. Another important property is that the determinant of a matrix is equal to the determinant of its transpose.

The exercise we're examining makes use of another property: when a scalar multiplies a row of a matrix, the determinant of the resulting matrix is equal to that scalar times the determinant of the original matrix. This is observed in the calculated determinants of both matrices. These properties are just a few among many, but they serve as the cornerstones for solving more intricate problems in matrix algebra.

2x2 Matrix Determinant

The determinant of a 2x2 matrix is found using a straightforward calculation. It's essential to understand this computation as it is often the first step in learning about determinants for larger matrices. To calculate the determinant of a 2x2 matrix, which is structured as

\begin{array}{cc} a & b \ c & d \-end{array}

, you take the product of the top left element \(a\) and the bottom right element \(d\), and subtract the product of the top right element \(b\) and the bottom left element \(c\). This process gives the formula

ad - bc.

In our original exercise, by applying this formula, we can see the determinant evaluated on the left matrix produces \(c(wz - xy)\), confirming the principle that multiplying a row by a scalar value \(c\) scales the determinant by that value. This is a particular instance of a more general principle in larger matrices, and it shows how integral the 2x2 determinant calculation is for understanding multidimensional spaces and transformations within linear algebra.