Determinant of 4x4 Matrix

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Introduction to Determinant of 4x4 Matrix

Determinant of a 4×4 matrix is a unique number that is also calculated using a particular formula. If a matrix order is in n x n, then it is a square matrix. So, here 4×4 is a square matrix that has four rows and four columns. If A is a square matrix then the determinant of the matrix A is represented as |A|.

 

Find out the determinant of 4×4 matrix? we will use the normal method, there is a determinant of 4×4 matrix formula which we normally use to find the determinant of a 3×3 matrix.

 

Definition of Determinant

The Easiest way will be to formulate the determinant by putting it into account of the top row of the elements and the corresponding minors. Take the first element of the top row and then multiply it by a minor, and after that subtract the product of the second element with the minor. Continue to alternately add and subtract the product of each element of the top row with its given minor until all the elements of the top row have been considered.

 

Determinants also play a very crucial role in finding the inverse of a matrix and also for solving the systems of the linear equations. In the following part we also assume that we have the square matrix (m equals n). The determinant of the matrix A will be put up by det(A) or |A|. First the determinant of a 2×2 and 3×3 matrix will be introduced, then the n×n case will be put on.


What is the Matrix?

Before learning the operations of the matrix, let us discuss what a matrix is. A matrix can be defined as the rectangular array of the numbers or the symbols which are normally arranged in rows and the columns. The order of the matrix also can be defined as the number of rows and no. of columns. The entries are also the numbers in the matrix and each of the numbers is called an element. The plural of the word matrix is known as matrices. 


The size of the matrix is referred to as ‘n by m’ matrix and it is written as m×n, where n is for the number of rows and m is for the number of columns. For example, we have a 3×2 matrix, that is because the number of rows here is equal to 3 and the number of columns here is equal to 2.

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The dimensions of the matrix also can be defined as the number of rows and columns of the matrix in that order. As the matrix A given above has 2 rows and 3 columns, so it is known as a 2×3 matrix.

 

Shortcut to Find the Rank of a Matrix

The total number of the linearly independent vectors in a matrix is the same as the whole total number of the non-zero rows in its row present in the echelon matrix. So, to find out the rank of a matrix, we have to generally transform the matrix to its row echelon form and then count the total number of non-zero rows.

 

Symbol of Determinant

The symbol for the determinant is two vertical lines on both sides.

Example:

|A| denotes the determinant of the matrix A

(the same symbol as the absolute value)


For a 2×2 Matrix

For a 2×2 matrix (2 rows , 2 columns):

A = \[\begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

The determinant is written as:

|A| = ad − bc


It is very easy to remember whenever you think of a cross:

  • Blue is taken positive (+ad),

  • Red is taken negative (−bc)


Determinant for a 2×2 Matrix

If A is an arbitrary 2×2 matrix A, the elements are given as:

A = \[\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\]

then the determinant of a and this matrix is put up as follows:

det(A) = lAl = \[\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\] = a\[_{11}\]a\[_{22}\] - a\[_{21}\]a\[_{12}\]


For a 3×3 Matrix

For a 3×3 matrix (3 rows , 3 columns):

A = \[\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]

The determinant is written as:

|A| will be equal to a(ei − fh) −  b(di − fg)  + c(dh − eg)

It may look difficult, but there is a known pattern:

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To work out the determinant for a 3×3 matrix here are the points:

  • Multiply a by the determinant of the 2×2 matrix which is not in a's row or the column.

  • Also for b, and for c too

  • Add them up, but also remember about the minus in front of the b

As for the formula (remember the vertical bars || denotes as"the determinant of"):

lAl = a . \[\begin{vmatrix} e & f \\ h & i \end{vmatrix}\] - b . \[\begin{vmatrix} d & f \\ g & i \end{vmatrix}\] + c . \[\begin{vmatrix} d & e \\ g & h \end{vmatrix}\]


How to Calculate the Determinant of 4×4 Matrix?

Before we try to find out the determinant of 4×4 matrix, first let us check a few conditions given below.

  • If there is any condition present , where the determinant of 4×4 matrix can be 0 (for example, the complete row or complete column is 0)

  • If factoring out any of the rows or the column is possible.

  • If the elements of the matrix are the same but then reordered on any of the columns or rows.

In any of the three cases given above is met, the corresponding methods for calculating 3 x 3 determinants are used. We change a row or a column to fill it up with 0, except for the one element. The determinant of the 4×4 matrix will be equivalent to the product of that element and its cofactor. In this situation, the cofactor is a 3×3 determinant which is estimated with its particular formula.

FAQ (Frequently Asked Questions)

Question 1: What Does the Determinant of a 4×4 Matrix of a Matrix Tell You?

Answer: The determinant of the 4×4 matrix is very important for solving the linear equations, by seeing how the linear transformation also changes the area or the volume, and changing variables in integrals. The determinant of a 4×4 matrix can also be viewed as the function whose input is a square matrix and its output is a number.

Question 2: What is the Meaning of Transpose of a Matrix?

Answer: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A equals (aᵢⱼ) and transpose of the A is:

AT equals (aᵢⱼ) where j is column number and i is row number of the matrix A.

Question 3: What are the Two Methods for Solving Matrices?

Answer: The  two methods used to solve matrices. These are:

  • Inverse Matrix Method

  • Cramer's Rule

Question 4: What is the Inverse Matrix Method?

Answer: The inverse matrix method uses the inverse of a matrix to help solve a system of equations, such like the above Ax = b. By multiplying before both of the sides of this equation by A-1 gives the value:

A⁻¹ (Ax) = A⁻¹ b

(A⁻¹ A)x = A⁻¹ b

or alternatively

X = A⁻¹ b