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In ‘Introduction to Determinants’ notes, we will understand the meaning of determinants and matrices and learn tricks to calculate the determinant of a matrix 3×3 and determinant of 2×2 matrix.
What is a matrix?
- The elements of a matrix are called entries.
- The matrix can be described as a rectangular array of numbers that are arranged in rows and columns.
- A matrix or matrices (plural) can be used to record data that depends on two parameters, describe a linear equation, or keep track of coefficients in a linear transformation.
- For example, a system of linear equation is-
p1x + q1y = c1 and p2x + q2y = c2 can be represented as
|p1 q1||x y|=|c1 c2|
|p2 q2|
- The concept of matrices is applied in various branches of mathematics, statistics, economics, physics, and engineering.
What is a determinant?
- In the English dictionary, a determinant refers to an element that determines or identifies the value or nature of something.
- In mathematics, every square matrix of order n can be associated with a real or complex number. That real or complex number is called the determinant of square matrix A.
- It is important to know that only square matrices can have determinants.
- For example, if X = p q r s , then the determinant of X will be denoted as X= p q r s =det X .
- Determinants are used to determine whether a system of n equations has a solution.
- There are several properties of determinants. Some important properties of determinants are-
- The value of the determinant doesn’t change even when the rows and columns of a matrix are interchanged.
- The sign of the determinant changes if any two rows or columns of that determinant are interchanged.
- The value of a determinant is zero if the elements of any two rows and columns are the same.
History of Determinants
- Many people believe that Japanese mathematician Seki Kowa was the first person to discover determinants. Seki Kowa, who was also called Japan’s Newton, was a mathematician and author during the Edo period. He is credited for laying the foundation for Japanese mathematics. In 1683, Kowa, in his book “Kai Fukudai no Ho,” showcased his idea of determinants.
- Around the same time, German Mathematician, Gottfried Leibniz, wrote about determinants as a way of solving linear equations in a letter to Guillaume de L’Hôpital.
- The theorems and properties for calculating determinants of matrices were discussed even further by other mathematicians in the following centuries.
- Colin Maclurin was an English Mathematician in the 18th century. His book “Treatise of Algebra” contained Cramer’s rule and indicated the ways of determining four simultaneous quadratic equations.
Determinant of a 1×1 Matrix
- The determinant of a 1×1 square matrix is simply the number within the matrix.
- For example, let X = [x] be a 1×1 square matrix. Then the determinant of X will be x itself.
Determinant of 2×2 Matrix
Let X be a 2×2 square matrix.
Det (X) = X = = (p × s) – (q × r )
For example, let us calculate the determinant of a matrix
Q = |2 3|
|5 9|
Det (Q) = Q =
= (2 × 9) – (3 × 5)= 18 – 15 = 3
Let us find the determinant of
matrix A =
Since, Det (A) = A = y2 – (y + 1) (y – 1) = y2 – (y2 – 1) = y2 – y2 + 1 = 1
Therefore, Det (A) = 1
Determinant of a Matrix 3×3
Let
X denotes a 3×3 matrix.
X=
The determinant of a matrix 3×3 can be calculated by either expanding its rows or columns. Therefore, there are six ways of calculating a 3×3. A determinant can be calculated across the 3 rows and 3 columns and always give the same result.
Expansion across the first row
The determinant of X can be calculated by expanding along X’s first row.
Step 1 in this process is to multiply p11 from Row 1 by (-1) (1+1) [ (-1)Sum of suffixes in p11] ,and the determinant is obtained by deleting the elements of the first row and the first column.
Therefore, p × (-1) (1+1) | t u|
| w x |
In Step 2, the second element of Row 1, q, is multiplied by (-1)1+2 and the determinant obtained by deleting the elements of row 1 and column 2. The resultant equation will look like- q × (-1)1+2 × |s u|
| v x|
In Step 3, the third element of row 1, r, is multiplied by (-1) 1+3 and with the determinant obtained by deleting the elements of the first row and the third column. The resultant equation is- r × (-1) 1+3 |s t |
|v w |
The Determinant of X = X = sum of all three resultant equations from steps 1, 2, and 3.
Therefore, X = (-1)2 p (t x – u w) + (-1)3 q (s x – u v) + (-1)4 r (s w – t v),
X= p t x – p u w – q s x + q u v + r s w – rt v… let this be equation 1.
Similarly, the determinant of X can be calculated across its second and third row. It should be noted that the value of the determinant will remain the same when it’s expanded across its first, second, and third row.
Expansion across column 1.
Similar to expansion across row 1, there will be three equations.
Equation 1 = p (-1)1+1
Equation 2 = s (-1)2+1
Equation 3 = v (-1)3+1
Therefore, X = p (-1)2
+ s(-1)3
+ v(-1)4
X = p (t x – u w) – s(q x – r w) + v (q u – r t),
= p t x – p u w – s q x + s r w + v q u – v r t… let this be equation 2.
We can observe that the value of the determinant, when calculated by expanding across row 1 and column 2 is the same.
Let us now, calculate the X=
Determining the value of X by expanding row 1,
X= 3 (56-20) – 4(63-10) + 5(18-8),
= 3 (36) – 4(53) + 5(10),
= 108 – 212 + 50 = -54
Determining the value of X by expanding column 2,
X = -4 (63 – 10) + 8 (21 – 5) – 2 (30 – 45),
= -4 (53) + 8 (16) – 2(-15),
= -212 + 128 + 30 = -54
The determinant of X will be defined as-
Determinant Matrix Calculator
- You can calculate the value of a determinant online with the help of an online determinant matrix calculator. Sites such as Matrix Reshish calculate the determinant of matrices of complex numbers for free.
- Such determinant matrix calculators can even calculate the determinant of non-square matrices and matrices which contain fractions and decimals.
Conclusion
With this, we conclude our article on introduction to determinants. Determinants are commonly categorized as- first order determinants, second order determinants, and third order determinants. In mathematics, the concept of determinants are used to provide a similar formula to find a solution of a system of n equations in n unknowns.
