Question

Explain why the determinant of the matrix is equal to zero. $$ \left[\begin{array}{rrrr} 3 & 4 & -2 & 7 \\ 1 & 3 & -1 & 2 \\ 0 & 5 & 7 & 1 \\ 1 & 3 & -1 & 2 \end{array}\right] $$

Short answer

The determinant of the given matrix is zero because it has two identical rows (rows 2 and 4).

Step-by-step solution

Identify Identical Rows

Examine the matrix and identify if there are any identical rows or columns. In the given matrix, Rows 2 and 4 have identical elements: \(1, 3, -1, 2\)

Apply the Property of Determinants

According to the properties of determinants, if two rows (or columns) of a square matrix are identical, then its determinant is zero. Since we have identified that rows 2 and 4 in the matrix are identical, we can apply this property.

Conclusion

As Rows 2 and 4 of the matrix are identical, according to the properties of determinants, the determinant of this matrix is zero.

Key concepts

Properties of Determinants

Understanding the properties of determinants is crucial when working with matrices in linear algebra. A determinant is a special number that can be calculated from a square matrix. It gives important information about the matrix, such as whether it has an inverse or the volume of a geometric figure associated with it. The determinant of a matrix adheres to several important properties which simplify the calculations and provide insights into the matrix's characteristics.

For instance, one of these properties is that if any two rows (or columns) of a matrix are identical, the determinant of that matrix is zero. This is because the determinant represents a signed volume in n-dimensional space, and two identical rows imply that the space is collapsed in one dimension; thus, there is no volume. Similarly, swapping two rows of a matrix will flip the sign of its determinant. Another property is that multiplying a row by a scalar multiplies the determinant by that scalar.

These properties can be applied to solve problems efficiently, like the one in the given exercise. The identification of identical rows allowed us to use the property that leads to a determinant of zero, avoiding lengthy computations that would otherwise be necessary using the standard method of cofactor expansion.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. The determinant is a fundamental concept in linear algebra because it gives insights into the solvability of a system of linear equations. Determinants can tell us if a system has a unique solution, infinite solutions, or no solution at all.

When the determinant of a matrix associated with a system of linear equations is zero, it often indicates that the system has either no solutions or an infinite number of solutions. This ties into the concept of row reduction and echelon forms, where the presence of a row of zeros would reflect these solution scenarios. Additionally, determinants play a key role in finding the inverse of a matrix, as a matrix is invertible only if its determinant is non-zero.

Determinants also show up in various areas like geometry, where they can describe areas, volumes, and related concepts, and in differential equations, where they are used in the method of Cramer's Rule to find solutions to systems of linear differential equations.

Square Matrix

A square matrix is a matrix with the same number of rows and columns, forming a perfect square. This type of matrix is central to many concepts in linear algebra, including determinants, as the determinant is only defined for square matrices.

Every element within a square matrix corresponds to two indices: the row and column it occupies. The positions in a matrix are used in various calculations, like finding the determinant, which involves the sum and product of elements based on their position. The square nature of these matrices also means they can represent linear transformations that keep the dimension of a vector space unchanged.

In the context of the original exercise, the presence of two identical rows within a square matrix directly influenced the determinant's value. This showcased an important aspect of how the structure of a square matrix, down to the equality or disparity between its rows and columns, can dramatically impact its properties and, by extension, the properties of the system or transformation it represents.