Just as the names of each of them sound, the general method is the "formal" method to use mathematically, following all the rules and producing some minor matrix determinant calculations along the way to find the final solution. There are two methods for finding the determinant of a 3x3 matrix: the general method and the shortcut method.
How to find the determinant of a 3x3 matrix Still, it is important to keep those properties in mind while performing the calculations of the exercises in the last section of this lesson.
Remember we will look at that complete topic in a later lesson called: properties of determinants. The lesson of today will be focused on the process to compute the determinant of a 3x3 matrix, taking approach of the matrix determinant properties, which have been briefly seen in past lessons. Notice the difference, the matrix is written down with rectangular brackets and the determinant of the matrix has its components surrounded by two straight lines. In other words, we usually write down matrices and their determinants in a very similar way:Įquation 1: Difference between the notation of a matrix and a determinant This last notation comes from the notation we directly apply to the matrix we are obtaining the determinant of. The determinant of a matrix can be denoted simply as det A, det(A) or |A|.
The determinant of a non square matrix does not exist, only determinants of square matrices are defined mathematically. Hence, the simplified definition is that the determinant is a value that can be computed from a square matrix to aid in the resolution of linear equation systems associated with such matrix. What is the determinant of a matrixīy using the knowledge that a matrix is an array containing the information of a linear transformation, and that this array can be conformed by the coefficients of each variable in an equation system, we can describe the function of a determinant: a determinant will scale the linear transformation from the matrix, it will allow us to obtain the inverse of the matrix (if there is one) and it will aid in the solution of systems of linear equations by producing conditions in which we can expect certain results or characteristics from the system (depending on the determinant and the type of linear system, we can know if we may expect a unique solution, more than one solution or none at all for the system).īut there is a condition to obtain a matrix determinant, the matrix must be a square matrix in order to calculate it. Knowing that, this lesson will focus on the process for evaluating the determinant of a 3x3 matrix and the two possible methods to employ. In such matrix, the results of each equation from the system will be placed on the right hand side of the vertical line which represents the equal sign. The matrix representation of a linear system is made by using all of the variable coefficients found in the system, and use them as element entries to construct the rectangular array of an appropriate size augmented matrix. In that way, we can resolve systems of linear equations by representing a linear system as a matrix. If you want to review the definition of the matrix with more detail you can revisit our lesson on notation of matrices.Ī matrix describes a linear transformation or linear map, which is a kind of transcription between two types of algebraic structures, such as vector fields.
This list can also be called a rectangular array, and it provides an orderly fashion to display a "list" of information elements. Remember that we have learnt that a matrix is an ordered list of numbers put in a rectangular bracket. The determinant of a 3x3 matrix (General and Shortcut method)Īs we have seen in past lessons, in order to define what is a determinant of a matrix we need to go back a to our definition of a matrix.