2 X 2 Matrix Determinant

In this lesson, we will acquire how to observe a determinant of a ii ten ii matrix. To find the determinant, we multiply the top left entry and bottom correct entry and subtract it with the production of the elevation right entry and lesser left entry. We volition use these determinants later on in the class to show if a matrix is invertible. We volition as well employ it to find inverses of 2 x 2 matrices.

The determinant of a 2x2 matrix

What is the determinant of a matrix

In gild to define what is a determinant of a matrix we need to go back a to our definition of a matrix. Remember that we accept learnt that a matrix is an ordered list of numbers put in a rectangular bracket. This listing can also exist called a rectangular array, and information technology provides an orderly fashion to display a list of data elements. If yous want to review the definition of the matrix with more detail you can revisit our lesson on notation of matrices.

A matrix describes a linear transformation or linear map, which is a kind of transcription betwixt two types of algebraic structures, such as vector fields. In that way, we can resolve systems of linear equations by representing a linear organization as a matrix. The matrix representation of a linear system is made by using all of the variable coefficients plant in the system, and apply them equally chemical element entries to construct the rectangular array of an appropriate size augmented matrix. In such matrix, the results of each equation from the system volition be placed on the right paw side of the vertical line which represents the equal sign.

You lot may exist asking yourself, what almost the definition of the determinant of a matrix so? Knowing that a matrix is an array containing the data of a linear transformation, and that this assortment can be conformed by the coefficients of each variable in an equation arrangement, we tin can describe the role of a determinant: a determinant will scale the linear transformation from the matrix, it will permit us to obtain the inverse of the matrix (if there is i) and it will aid in the solution of systems of linear equations past producing conditions in which we can expect certain results or characteristics from the system (depending on the determinant and the type of linear organisation, we can know if nosotros may look a unique solution, more than than i solution or none at all for the arrangement).

Simply there is a status to obtain a matrix determinant, the matrix must be a square matrix in order to calculate it. Hence, the simplified definition is that the determinant of nxn matrix is a value that can be computed from a square matrix to assist in the resolution of linear equation systems associated with such matrix. The determinant of a non square matrix does not exist, only determinants of foursquare matrices are defined mathematically.

The determinant of a matrix can be denoted simply every bit det $A$ , det( $A$ ) or $\mid A\mid$ . This last notation comes from the notation we directly apply to the matrix we are obtaining the determinant of. In other words, nosotros usually write down matrices and their determinants in a very similar fashion:

Equation 1: Divergence between the note of a matrix and a determinant

But notice, when we write down the matrix we have ii brackets that are closed in, and when we are referring to the determinant of the matrix, the matrix components are surrounded by 2 directly lines.
Nosotros volition talk more nearly this topic on a later lesson dedicated to the properties of determinants , notwithstanding, we will introduce a few of these backdrop quickly today today so you tin have a meliorate idea on how to observe the determinant of a 2x2 matrix and why is this important.

How to find the determinant of a matrix

The computation of the determinant of a matrix varies depending on the dimensions of the matrix itself, as the dimensions of the matrix increase, the determinant calculation gets extensive and more complicated.
On this lesson we will focus on the calculation of determinants from 2x2 square matrices but, we will study the steps to obtain the determinant of a 3x3 matrix (Full general & Shortcut method) in a later lesson.

How do y'all detect the determinant of a 2x2 matrix

Defining a second order matrix $X$ as:

Equation 2: Matrix X

Its determinant is mathematically defined to exist:

$det (X) = advert - bc$

Equation 3: Determinant of matrix 10

Which can besides be written as:

Equation 4: Determinant of matrix X in rectangular array class

The only simpler determinant to obtain also the determinant of a 2x2 matrix is the determinant of 1x1 matrix which is defined to be equal to the only element within the matrix.

Determinant of triangular matrix

A triangular matrix, as the proper noun suggests, is that which has its chemical element entries grade a triangle of some sort by having all zeros in a corner either beneath or higher up the main diagonal. Thus, there are two types of triangular matrices: upper triangular matrices and lower triangular matrices.
An upper triangular matrix is a square matrix in row echelon course in which all of the pivots are in consecutive columns producing the master diagonal, thus resulting in a triangle made out of zeros below the chief diagonal in the left bottom corner. A lower triangular matrix has the positions of the zeros in reverse position from the upper triangular matrix, therefore, a lower triangular matrix is that where the zeros are constitute in a higher place the main diagonal to the upper right corner.

Equation 5 :Upper and lower triangular matrices

Given that but the upper triangular matrix is in echelon grade, which is essential when solving systems of linear equations through matrices and row operations, that is the triangular matrix which we are interested in.

It and then happens that the determinant of an upper triangular matrix is the outcome from the multiplication of all of the elements found in its diagonal, no matter the social club (size) of the matrix. Therefore, when row reducing a square matrix formed by a system of linear equations, if y'all arrive to an echelon form that is equal to an upper triangular matrix, you tin can hands compute its determinant and through that, obtain information virtually the kind of solutions to be expected from the organization.

Determinant of a diagonal matrix

A diagonal matrix is that in which all of its elements, except those in the diagonal, are zeros.

Equation 6: General definition of a diagonal matrix

In general, the determinant of a diagonal matrix is equal to the product of the entries on its diagonal. If we think of determinant $X$ found in equation ii as a diagonal matrix, that means that the elements $b$ and $c$ are equal to zero, therefore, nosotros can prove that the determinant of such diagonal matrix would exist the result of multiplying the entries on the matrix diagonal by following equation 3 to summate it. The result of calculating the 2x2 matrix determinant of $X$ in this case would be equal to (a)(d)-(0)(0)=ad, which is simply, multiplying the diagonal elements.

Notice that to obtain the determinant of a triangular matrix and a diagonal matrix nosotros can utilize the same uncomplicated approach. If you think about it, this comes from the fact that in a diagonal matrix, all of the components above and below the diagonal are zeros, thus if we follow the formula for the determinant of a 2x2 matrix shown in equation 3, nosotros volition see that the 2nd term in the formula is a multiplication of 2 zeros giving a zero to subtract from those elements n the diagonal and thus leaving them as the only result. Something very like happens in triangular matrices, only instead of having a multiplication of two zeros in the 2nd term of the determinant formula, you have the multiplication of a zero and a non-zero value, which nevertheless results in zero and leaves the kickoff multiplication every bit result.

Disruptive? Have a look at the next equations for each of the cases mentioned:

Equation 7: Determinant of diagonal and triangular matrices

Determinant of symmetric matrix

A symmetric matrix is a square matrix which happens to be identical to its transpose, and so, what is the transpose of a matrix?

Nosotros obtain the transpose of a matrix past rearranging the columns of the original matrix as rows in the transpose. For the case of square matrices, the transpose matrix will remain to be the same guild since information technology will keep to have the same corporeality of rows and columns than the original.
The process of obtaining the transpose of a matrix tin be seen in the next equation:

Equation 8: Obtaining the transpose of a 2x2 matrix

For the case of a third caste square matrix, the transpose can be obtained in the aforementioned fashion, just equally seen below:

Equation ix: Obtaining the transpose of a 3x3 matrix

Equally you tin run into, if symmetric matrix means a given matrix is equal to its transpose, information technology likewise means the symmetry, which comes from respect to the diagonal of the matrices, volition let all the elements inside the matrix to remain equal. And then, the determinant of a symmetric matrix, no thing its size, is calculated using the same procedure equally with whatsoever other foursquare matrices, only in that location is a special instance: all diagonal matrices are too, symmetric matrices, and then, their determinants tin be calculated just by multiplying its diagonal elements.

Matrix determinant backdrop

Besides finding the determinant of a matrix, in that location are interesting characteristics and conditions associated with this value (the determinant issue) that provide useful information about operations and processes that tin exist calculated with matrices, such as matrix inversion, diagonalization and the result of certain row operations on the determinant results (which at the same time, may or may not affect invertibility, etc

The interesting characteristics and conditions coming from finding the determinant of a 2x2 matrix (or any other square matrix of higher guild for that matter) are what we call backdrop of the result. Although such properties will be covered in detail in a afterwards lesson (every bit mentioned earlier), let us requite at least a brief caption of them and the types of operations they are associated with in relation to the matrix.

The backdrop of determinants are categorized in three sections each depending to the type of matrix operations they are associated with:

1. Row operation backdrop
2. Multiplicative properties
3. Other miscellaneous properties

Row operation properties of a 2x2 matrix determinant are associated, as their name suggests, to the iii types of matrix row operations that can be performed while row reducing a matrix, and how they will affect the upshot of the determinant of the matrix.

The main three properties related to row operations are:

1. If a multiple of one row of a given matrix is added to another row to produce a second matrix, which retains the aforementioned proportionality as the first, and then the determinant of the original matrix is equal to the determinant of the 2nd matrix obtained after the row performance.
2. If ii rows are interchanged in a matrix, the determinant of the original matrix is equal to the negative of the determinant of the resulting matrix afterward the row substitution.
3. If a constant is multiplied to a row within a given matrix which produces a different second matrix, the determinant of resulting matrix after the multiplication of the constant is equal to the determinant of the original matrix times the same constant.

The main multiplicative property determinants is probably that if you multiply 2 matrices and and so obtain the determinant of the resulting one, this determinant value would exist equal if you were to obtain the determinant of each matrix separately and then multiply them with each other. Only said: $det (AB) = det(A) x det(B)$ .

Other miscellaneous properties talk about how the determinant of a matrix is equal to the determinant of its transpose, and one of the most important ones: that a matrix is invertible if, and just if, its determinant is NOT aught. This last property is of main importance for our next lesson where nosotros will innovate matrix inversion.

Think, all of the properties and rules we have seen today, although we accept focused in 2x2 matrices, they do utilise equally when finding the determinant of a large matrix. Also accept into account all of the properties have been just briefly discussed as an introduction for this lesson, for extensive explanations on them please visit our lesson almost properties of determinants, which comes later in our course.

Finding the determinant of a 2x2 matrix

At present that we have seen the general method of how to find the determinant of a 2x2 matrix and that we accept seen different matrix cases and backdrop, is fourth dimension for usa to utilize the knowledge obtained and solve a few example exercises:

Example exercises

1. $\quad$ We define the 2x2 matrix shown below as $X$ :

Equation 10: Matrix X

Find the determinant of matrix $X$ by following the equation for the determinant of a 2x2 matrix shown in the 2d department of this lesson:

$det (X)=(46)-(71)=24-vii=17$

Equation eleven: Determinant of matrix X

2. $\quad$ Given the 2x2 matrix A every bit shown below

Equation 12: Matrix A

Observe the determinant of 2x2 matrix $A$ :

$det (A)=(-24)-(ane-1)=-8+one=-7$

Equation 13: Determinant of matrix A

3. $\quad$ Given the 2x2 matrix E as shown below

Equation xiv: Matrix Eastward

Evaluate the determinant of the matrix Due east:

$det (E)=(-eight-11)-(-nine-4)=88-36=52$

Equation fifteen: Determinant of matrix E

4. $\quad$ Given the 2x2 matrix F as shown below

Equation xvi: Matrix F

Find its determinant:

$det (F)=(two-v)-(-76)=-10-(-42)=32$

Equation 17: Determinant of matrix F

five. $\quad$ Given the 2x2 matrix Y as shown beneath

Equation eighteen: Matrix Y

Notice its determinant:

$det (Y)=(-9-4)-(215)=36-xxx=half dozen$

Equation 19: Determinant of matrix Y

half dozen. $\quad$ Given the 2x2 matrix Z as shown beneath

Equation 20: Matrix Z

Find its determinant:

$det (Z)=(3-4)-(510)=-12-50=-62$

Equation 21: Determinant of matrix Z

***

Look at the values obtained throughout all of the six exercises done above on the determinant of a 2x2 matrix, even for a minor matrix of dimensions 2x2, these values are significant in finding if such matrices can, or cannot, exist inverted. Our next section will exist focused on this, the 2x2 invertible matrix, or in other words, how exercise nosotros know if we can invert a matrix.

We hope this lesson has been useful and of your enjoyment, before we jump onto the next one let u.s.a. requite you a few recommendations so you tin can go along your studies on the topic of today. First, nosotros recommend yous to visit this brief lesson on determinants, where you will find an introduction to a higher order determinant at the terminate of information technology, and so you tin can take a look into this other lesson which covers both determinants and Cramer'due south rule. Finally, we would like to recommend this article on the the determinant of a square matrix which funnily enough, mentions at the beginning that the author has not been able to find a good English language definition of what a determinant is (at least at the time he wrote the text). We promise this StudyPug lesson has been able to provide that for sure: a clear definition of the determinant of a square matrix (but if not, practise not hesitate to ask!).

Well, this has been fun today, see you onto the next lesson!

2 X 2 Matrix Determinant,

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