Determinants are considered an algebraic representation of the sum of the products of elements each with an algebraic symbol, typically in a square array and used for the solution of systems of linear equations.
The determinant for linear algebra is a scalar value that can be determined by square matrix elements and which encodes those linear transformation properties defined in a matrix. The determinant of a matrix A is denoted det(A), det A, or |A|.
The volume scaling factor defined in the matrix can be considered geometrically as the linear transformation. This is also the signed n-dimensional volume occupied by the matrix ‘ column or row vectors. The determinant is either positive or negative based on the preservation or reversal of n-space orientation through the linear mapping.
In the case of a 2 × 2 matrix the determinant may be defined as:
In the cast of 3×3 matrix, the determinant may be defined as:
Now, it is time to go through the types of determinant matrixes.
What is a 2×2 matrix?
A matrix that contains two rows and two columns is known as 2x2matrix. A 2X2 matrix is a tool used to help gain insight and outcomes in a dialogue. On each end of the spectrum designers create a matrix of 2×2 with opposite features (i.e. cheap versus costly).
Why to use a 2×2 matrix?
A 2×2 matrix is a tool that allows people to think and talk about issues. Use it to help you learn about connections between things or people during your synthesis process. It is expected that a 2×2 will provide input. 2×2 matrices are perfect for expressing a reference you want to communicate visually.
How to use a 2×2 matrix?
For a 2×2 matrix (2 rows and 2 columns):
The determinant is:
|A| = ad − bc
“The determinant of A equals a times d minus b times c”
If the input in the matrix is real, Matrix A can be used in two linear maps: One, which maps the regular base vectors to A rows, and the other, which maps them to A columns- A. In other words, you subtract the top-to-bottom-right diagonal to take the determinants of a 2-2 matrix, from which you extract the product from the bottom-left-to-top-right diagonal.
It is easy to remember when you think of a cross:
- Blue is positive (+ad),
- Red is negative (−bc)
Find determinant of the following matrix ‘B’:
|B|= 4×8 − 6×3
Inverse of a 2×2 matrix:
So, how to calculate the inverse of a 2×2 matrix?
It’s very easy. For a 2×2 matrix:
In other words; switch the positions of a and d, put negatives on b and c, and split all by evaluating (ad-bc). Just for example:
For your kind information:
it must be true that: A × A-1 = I
Multiplication of a 2×2 matices:
That row of the first matrix is taken and every column of the second matrix is multiplied. Together, that contributes. And after that, find the determinant of that resulted matrix.
A matrix that contains three rows and three columns is known as 3×3 matrix.
How to find/solve a 3×3 matrix?
The traditional method for calculating a matrix of 3×3 is a breakdown of smaller, easy-to-manage, evaluating problems of 2×2.
For a 3×3 matrix:
Inverse of a 3×3 matrix:
Here, we have used Elementary Row operation for finding the inverse of a 3×3 matrix.
We begin with matrix A and write it down next to it with an identity matrix I:
(This is called the “Augmented Matrix”)
Here, ’I’ is known as Identity Matrix.
The “Identity Matrix” is the matrix equivalent of the number “1”:
A 3×3 Identity Matrix
- It is “square” (has same number of rows as columns),
- It has 1s on the diagonal and 0s everywhere else.
- It’s symbol is the capital letter I.
But in Elementary Row Operation, we can only do,
- swap rows
- multiply or divide each element in a a row by a constant
- replace a row by adding or subtracting a multiple of another row to it
We do these steps as follows:
No.1- Start with A next to I
No.2- Add row 2 to row 1,
No.3- then divide row 1 by 5,
No.4- Then take 2 times the first row, and subtract it from the second row,
No.5- Multiply the second row by -1/2,
No.6- Now swap the second and third row,
No.7- Last, subtract the third row from the second row,
And we are done!
And matrix A has been made into an Identity Matrix and at the same time an Identity Matrix got made into A-1.
Multiplication of 3×3 matrix:
The following example explains the multiplication of two 3×3 matixes.
Then find the determinant of the resulted matrix ‘C’, by using the method that is explained earlier.
A matrix having four rows and four columns is known as 4×4 matrix. E.g:this is a 4×4 matrix.
The inverse of a 4×4 matrix:
The following is the easiest formula that shows the way of finding the inverse of a 4×4 matrix;
Using this formula, you can easily find the determinant of a 4×4 matrix by applying the determinant formula to the resulted matrix.
Applications of determinants:
Following are the applications of a determinant:
- Linear Independence
- The orientation of a basis
- Volume and Jacobean Determinant
- Vandermonde Determinant