Why find determinants

It can be proven that any matrix has a unique inverse if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.

In the case where the matrix entries are written out in full, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the brackets or parentheses of the matrix. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, shown below:.

In linear algebra, the cofactor sometimes called adjunct describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. To know what the signed minor is, we need to know what the minor of a matrix is. Google Classroom Facebook Twitter. Video transcript As a hint, I will take the determinant of another 3 by 3 matrix. But it's the exact same process for the 3 by 3 matrix that you're trying to find the determinant of.

So here is matrix A. Here, it's these digits. This is a 3 by 3 matrix. And now let's evaluate its determinant. So what we have to remember is a checkerboard pattern when we think of 3 by 3 matrices: positive, negative, positive. So first we're going to take positive 1 times 4. So we could just write plus 4 times 4, the determinant of 4 submatrix.

And when you say, what's the submatrix? Well, get rid of the column for that digit, and the row, and then the submatrix is what's left over. So we'll take the determinant of its submatrix. So it's 5, 3, 0, 0. Then we move on to the second item in this row, in this top row. But the checkerboard pattern says we're going to take the negative of it. So it's going to be negative of negative let me do that in a slightly different color-- of negative 1 times the determinant of its submatrix.

The process of forming this sum of products is called expansion by a given row or column. Now find the cofactor of each of these minors.

The determinant is found by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products. Work carefully, writing down each step as in the examples. Skipping steps frequently leads to errors in these computations. Exactly the same answer would be found using any row or column of the ma trix.

One reason that column 2 was used in Example 3 is that it contains a 0 element, so that it was not really necessary to calculate M 32 and A 32 above. One learns quickly that zeros can be very useful in working with determinants. Thus, these arrays of signs can be reproduced as needed.

Knowing that, this lesson will focus on the process for evaluating the determinant of a 3x3 matrix and the two possible methods to employ. By 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.

But there is a condition 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 is a value that can be computed from a square matrix to aid in the resolution of linear equation systems associated with such matrix. The determinant of a non square matrix does not exist, only determinants of square matrices are defined mathematically.

The determinant of a matrix can be denoted simply as det A, det A or A. This last notation comes from the notation we directly apply to the matrix we are obtaining the determinant of. In other words, we usually write down matrices and their determinants in a very similar way:. 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.

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.

Remember we will look at that complete topic in a later lesson called: properties of determinants. Still, it is important to keep those properties in mind while performing the calculations of the exercises in the last section of this lesson. There are two methods for finding the determinant of a 3x3 matrix: the general method and the shortcut method.

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. While the shortcut method is more of a clever trick we can use to simplify the calculation, still being careful to not forget numbers, the order in which they have to be multiplied and some rearrangements of the elements in the matrix.

After you take a look at both methods to find the determinant of a 3x3 matrix, you can always pick whichever suits you best and use it for your studies, but remember that it is important you know both of them in case you are ever asked to compare results from them. So, without further delay let us define the determinant of 3x3 matrix A as shown below, so we can observe how it can be calculated through both methods:.

The general method to obtain the determinant of a 3x3 matrix consists of breaking down the matrix into secondary matrices of smaller dimensions in a process called "expansion of the first row". What this process does is to use the elements from the first row of the 3x3 matrix and use them as factors in a sum of multiplications where the big matrix gets redistributed. Taking as a reference the 3x3 matrix determinant shown in equation 2, we construct the first part of the result of this operation by selecting the first element of the first row and column which is constant "a" , and then multiply it by a matrix produced from the four elements which do not belong to either the row of the column in which "a" is.

Multiply "a" with this secondary 2x2 matrix obtained and that is the first term of the solution. The process is called an expansion of the first row because as you can see in equation 3, all of the elements from the first row of the original 3x3 matrix remain as main factors in the expansion to be solved for. All of the 2x2 matrices in the expansion are what we call "secondary matrices", and they can be easily resolved using the equation learnt on the determinant of a 2x2 matrix lesson.

And so, taking into consideration the formula for the determinant of a square matrix with dimensions 2x2, we can see that equation 3 yields: Equation 4: Equation for determinant of a 3x3 matrix through general method part 2. At this point you may have noticed that finding the determinant of a matrix larger than 2x2 becomes a long ordeal, but the logic behind the process remains the same and so the difficulty is similar, the only key point is to keep track of the operations you are working through, even more with even larger matrices than a 3x3.

The determinant of a 3x3 matrix shortcut method is a clever trick which facilitates the computation of a determinant of a large matrix by directly multiplying and adding or subtracting all of the elements in their necessary fashion, without having to pass through the matrix expansion of the first row and without having to evaluate secondary matrices' determinants.

The whole process of how to evaluate the determinant of a 3x3 matrix using the shortcut method can be seen in the equation below:. When computing the determinant of an nxn matrix in this case a 3x3 matrix as shown above, notice we first rewrite the matrix accompanied by a repetition of its two first columns now written outside to the right hand side. Then, the determinant value will be the result of the subtraction between addition of products from all of the down-rightward multiplications and the down-leftward multiplications.

Said clearer, there will be a total of three complete diagonals going from the top left to the bottom right, and another set of three complete diagonals going from the top right to the bottom left. We will multiply the elements of each diagonal together, then add them with the results coming from the other diagonals. There is something to have in mind, all of the diagonals' multiplications going from top left to bottom right have an intrinsic positive sign multiplied to them, while all of the diagonals' multiplications going from top right to bottom left have an intrinsic negative sign multiplied to them, and so, when adding the results from all of the multiplications, a subtraction such as the one shown in equation 5 will result.