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Determinants are a fundamental part of linear algebra. They’re an important concept because they make the notation elegant and flexible, more concrete. They also allow us to explore the deep properties of matrices in a very intuitive fashion. Determinants are pretty simple to handle, but it’s important to know when you should use them and when you shouldn’t. Determinants of a matrix can be defined as a determinant of its transpose. In other words, the determinant of a matrix A is the same as the determinant of its transpose AT. This article is a rundown of the basics of determinants.
About determinants:
Suppose we have an n by n matrix A. The determinant, then, is a single number associated with each element in the matrix; specifically, it’s associated with each element in row i and column j. It turns out that the determinant of an element is given by the product of the elements in its two adjacent rows and two adjacent columns.
Let’s take a concrete example with two by two matrices. Let A be the matrix with 1s along the diagonal and 0s elsewhere. A determinant is just a single number that tells us a lot about this matrix.
Here’s a quick example:
If we put the matrix on a table and then lift the left side, which is the first row of coefficients and then lift the right side, which is the second row of coefficients, what’s left under those rows?
It must be zero because if those two rows are lifted to make a column, and that column is called A again, then what’s under A has to be zero. That means that each element of A has to be equal to its cofactor.
A determinant of a square matrix A is a single number, denoted by det “A,” which satisfies the equation
det “A” = det “AT.” Here, “T” denotes the transpose operation, and A defines the determinant of a matrix. This section will explore how we can calculate determinants in terms of elementary row operations on the matrix.
Like any number, one can define its inverse. On the left is the determinant, and on the right is its inverse.
The determinant of a square matrix A is a special function whose value depends on how you multiply and transpose an array of numbers (called the elements of A) that appear in a certain order — like an array of equations. Here’s how it works: Let’s say we’re given an array with three rows and three columns, or 3×3:
We’ll denote this array by “a.”
How determinants are used in linear algebra:
The most obvious use of determinants is to calculate whether or not a matrix will lead to a nontrivial solution when used to solve a system of linear equations.
It is quite useful, especially if the system being solved is particularly complex (or the determinant of the coefficient matrix is large).
Determinants are also calculated using a bit of algebra with the transpose operation on an array of numbers. They’re more concrete than matrices because this method is easy to visualise and understand. By knowing how to calculate the determinant, we can explore the real world: calculating an array’s determinant can be used to prove or disprove whether or not it complies with some matrix equation.
Properties of a matrix:
A square matrix is a matrix that has the same number of rows as columns. For instance, the following matrices are square: 2×2, 3×3, 4×4, etc.
A non-square matrix is a matrix that is not square. For example, at least one column with all zeros exists for every row in the matrices.
If you multiply two matrices using matrix multiplication, it doesn’t matter which order you do it in, as long as you don’t switch the order of any rows or columns.
In other words, two matrices, A and B, can be multiplied in any order, and it turns out that their determinant will always be the same.
Switching property:
Let’s take a concrete example with two by two matrices. Let A be the matrix with 1s along the diagonal and 0s elsewhere. A determinant is just a single number that tells us a lot about this matrix.
The switching property of determinants says that if two different matrices have the same determinant, they must have the same number of rows and columns.
Take another concrete example of the switching property with two by two matrices. Let A and B be two by two matrices with 1s along the diagonal and 0s everywhere else. If we calculate the determinant, we should find that it will be 17.
Conclusion:
As one can see, a determinant is used to help calculate nontrivial solutions for linear equations because the determinant of a matrix depends on the solution of the system and will be 0 if and only if some of the original equations are linearly dependent. We need to define a special matrix called a “cofactor matrix to calculate such quantities.” A cofactor matrix is just a modified version of the original matrix, in which factors that multiply a determinant are replaced by their cofactors. The switching property of determinants is used in multiple fields of algebra.
